| L(s) = 1 | + 4·3-s + 8·9-s + 8·11-s + 8·13-s − 4·25-s + 20·27-s + 32·33-s − 8·37-s + 32·39-s − 8·47-s + 12·49-s + 40·59-s − 48·61-s − 8·71-s + 8·73-s − 16·75-s + 50·81-s + 24·83-s − 24·97-s + 64·99-s + 8·107-s − 32·109-s − 32·111-s + 64·117-s + 36·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s + 2.41·11-s + 2.21·13-s − 4/5·25-s + 3.84·27-s + 5.57·33-s − 1.31·37-s + 5.12·39-s − 1.16·47-s + 12/7·49-s + 5.20·59-s − 6.14·61-s − 0.949·71-s + 0.936·73-s − 1.84·75-s + 50/9·81-s + 2.63·83-s − 2.43·97-s + 6.43·99-s + 0.773·107-s − 3.06·109-s − 3.03·111-s + 5.91·117-s + 3.27·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8039105753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8039105753\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 11 | \( ( 1 - T )^{8} \) |
| good | 7 | \( ( 1 - 6 T^{2} + 62 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 4 T + 30 T^{2} - 124 T^{3} + 446 T^{4} - 124 p T^{5} + 30 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 76 T^{2} + 2880 T^{4} - 73652 T^{6} + 1419278 T^{8} - 73652 p^{2} T^{10} + 2880 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 - 64 T^{2} + 2532 T^{4} - 70080 T^{6} + 1500646 T^{8} - 70080 p^{2} T^{10} + 2532 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 + 20 T^{2} + 1078 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 144 T^{2} + 9252 T^{4} - 370480 T^{6} + 11535846 T^{8} - 370480 p^{2} T^{10} + 9252 p^{4} T^{12} - 144 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( 1 - 88 T^{2} + 5436 T^{4} - 233448 T^{6} + 8313094 T^{8} - 233448 p^{2} T^{10} + 5436 p^{4} T^{12} - 88 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 4 T + 16 T^{2} - 308 T^{3} - 1570 T^{4} - 308 p T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 208 T^{2} + 22596 T^{4} - 1574768 T^{6} + 76695494 T^{8} - 1574768 p^{2} T^{10} + 22596 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 - 92 T^{2} + 7712 T^{4} - 358404 T^{6} + 18505454 T^{8} - 358404 p^{2} T^{10} + 7712 p^{4} T^{12} - 92 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 4 T + 112 T^{2} + 484 T^{3} + 6990 T^{4} + 484 p T^{5} + 112 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 224 T^{2} + 26940 T^{4} - 2215968 T^{6} + 134459878 T^{8} - 2215968 p^{2} T^{10} + 26940 p^{4} T^{12} - 224 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 20 T + 304 T^{2} - 3300 T^{3} + 28766 T^{4} - 3300 p T^{5} + 304 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 24 T + 388 T^{2} + 72 p T^{3} + 39814 T^{4} + 72 p^{2} T^{5} + 388 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 180 T^{2} + 16358 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 4 T + 192 T^{2} + 900 T^{3} + 18126 T^{4} + 900 p T^{5} + 192 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 4 T + 198 T^{2} - 780 T^{3} + 19726 T^{4} - 780 p T^{5} + 198 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 32 T^{2} + 8516 T^{4} - 1034592 T^{6} + 38743814 T^{8} - 1034592 p^{2} T^{10} + 8516 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 12 T + 286 T^{2} - 2596 T^{3} + 34654 T^{4} - 2596 p T^{5} + 286 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 136 T^{2} + 25692 T^{4} + 2489400 T^{6} + 301060486 T^{8} + 2489400 p^{2} T^{10} + 25692 p^{4} T^{12} + 136 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 12 T + 264 T^{2} + 3252 T^{3} + 32318 T^{4} + 3252 p T^{5} + 264 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.60887741591621394528575819720, −3.56727050215389541491425248435, −3.56539029586008747845441617920, −3.47060840833850225053988562604, −3.34814571815109552061906603867, −3.25338804344753079490117985267, −2.80770708155705491832091345992, −2.77777144248681119239888947662, −2.65558077663096870785313985636, −2.64423274114393724540819916823, −2.46591234873333198794157039062, −2.39946020080705821675538962627, −2.31979815089846440384069839655, −2.18252321682107126988697016184, −1.95487290699384287984091680346, −1.66409466153467289599224233821, −1.43433019460739867108112352923, −1.42238565437100527085902475495, −1.38849016622675551008670651959, −1.36762611085550147938262704886, −1.17212171926409555077606734972, −1.00687048314697973099187019604, −0.76104904999974834715801243113, −0.43867976525762619747633127712, −0.03568918302190864357438898039,
0.03568918302190864357438898039, 0.43867976525762619747633127712, 0.76104904999974834715801243113, 1.00687048314697973099187019604, 1.17212171926409555077606734972, 1.36762611085550147938262704886, 1.38849016622675551008670651959, 1.42238565437100527085902475495, 1.43433019460739867108112352923, 1.66409466153467289599224233821, 1.95487290699384287984091680346, 2.18252321682107126988697016184, 2.31979815089846440384069839655, 2.39946020080705821675538962627, 2.46591234873333198794157039062, 2.64423274114393724540819916823, 2.65558077663096870785313985636, 2.77777144248681119239888947662, 2.80770708155705491832091345992, 3.25338804344753079490117985267, 3.34814571815109552061906603867, 3.47060840833850225053988562604, 3.56539029586008747845441617920, 3.56727050215389541491425248435, 3.60887741591621394528575819720
Plot not available for L-functions of degree greater than 10.
