| L(s) = 1 | + 4·4-s + 10·16-s − 64·47-s + 36·64-s − 32·67-s + 12·81-s − 64·89-s + 96·101-s + 32·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 96·169-s + 173-s + 179-s + 181-s − 256·188-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2·4-s + 5/2·16-s − 9.33·47-s + 9/2·64-s − 3.90·67-s + 4/3·81-s − 6.78·89-s + 9.55·101-s + 3.15·103-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 7.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 18.6·188-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.883276929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.883276929\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( ( 1 - p T^{2} + T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 3 | \( 1 - 4 p T^{4} + 70 T^{8} - 4 p^{5} T^{12} + p^{8} T^{16} \) |
| 5 | \( 1 + 32 T^{4} + 994 T^{8} + 32 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( 1 + 20 T^{4} + 1702 T^{8} + 20 p^{4} T^{12} + p^{8} T^{16} \) |
| 11 | \( 1 + 180 T^{4} + 27014 T^{8} + 180 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 52 T^{2} + 1270 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 748 T^{4} + 677926 T^{8} - 748 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( 1 + 1440 T^{4} + 1029794 T^{8} + 1440 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( 1 - 1196 T^{4} + 452454 T^{8} - 1196 p^{4} T^{12} + p^{8} T^{16} \) |
| 37 | \( 1 - 1824 T^{4} + 1700066 T^{8} - 1824 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 96 T^{2} + 4608 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )( 1 + 96 T^{2} + 4608 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 43 | \( ( 1 - 148 T^{2} + 9046 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 104 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( 1 - 2016 T^{4} + 2787746 T^{8} - 2016 p^{4} T^{12} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 8 T + 142 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( 1 - 6636 T^{4} + 26552486 T^{8} - 6636 p^{4} T^{12} + p^{8} T^{16} \) |
| 73 | \( 1 - 7104 T^{4} + 25157954 T^{8} - 7104 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( 1 + 13140 T^{4} + 120455654 T^{8} + 13140 p^{4} T^{12} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 196 T^{2} + 18774 T^{4} - 196 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 16 T + 240 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 6720 T^{4} + 127407362 T^{8} - 6720 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−5.30437005799406043053290783493, −5.03812911866289339728221104248, −4.89997935681123598356019102567, −4.87902954520553850735156964932, −4.67722716815156011943295611576, −4.66607511876759757033411579124, −4.58696836252256286494051540369, −4.45932276436874365070804502745, −3.83046329924954175392518814145, −3.69110293371478897580819856708, −3.55749628503680677566063466903, −3.54116415336237961114641433409, −3.47824574219652293956171773440, −3.30521527029069712957011117354, −2.91451810997620976672186583803, −2.80465036338089220196351953671, −2.79505963918029664180681095437, −2.39004888137066319267802666097, −2.14545997283557492071783253481, −2.11509405830754624110320446597, −1.63162276623333711169634711212, −1.62928271814679425847000629181, −1.37382657666514081550385711086, −1.22741314662488303593048439230, −0.29145372465863684917749713095,
0.29145372465863684917749713095, 1.22741314662488303593048439230, 1.37382657666514081550385711086, 1.62928271814679425847000629181, 1.63162276623333711169634711212, 2.11509405830754624110320446597, 2.14545997283557492071783253481, 2.39004888137066319267802666097, 2.79505963918029664180681095437, 2.80465036338089220196351953671, 2.91451810997620976672186583803, 3.30521527029069712957011117354, 3.47824574219652293956171773440, 3.54116415336237961114641433409, 3.55749628503680677566063466903, 3.69110293371478897580819856708, 3.83046329924954175392518814145, 4.45932276436874365070804502745, 4.58696836252256286494051540369, 4.66607511876759757033411579124, 4.67722716815156011943295611576, 4.87902954520553850735156964932, 4.89997935681123598356019102567, 5.03812911866289339728221104248, 5.30437005799406043053290783493
Plot not available for L-functions of degree greater than 10.
