| L(s) = 1 | + 4·11-s + 40·19-s + 12·29-s − 40·71-s + 76·79-s + 81-s − 68·109-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 160·209-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 9.17·19-s + 2.22·29-s − 4.74·71-s + 8.55·79-s + 1/9·81-s − 6.51·109-s + 8/11·121-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 + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 11.0·209-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.30758963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.30758963\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 3 | \( 1 - T^{4} - 95 T^{8} - p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( 1 - 41 T^{4} - 495 T^{8} - 41 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( 1 + 46 T^{4} + 24051 T^{8} + 46 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( 1 - 601 T^{4} + 236505 T^{8} - 601 p^{4} T^{12} + p^{8} T^{16} \) |
| 19 | \( ( 1 - 5 T + p T^{2} )^{8} \) |
| 23 | \( 1 + 287 T^{4} + 484593 T^{8} + 287 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 3 T + 13 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( 1 - 1106 T^{4} + 1569075 T^{8} - 1106 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 57 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( 1 - 5084 T^{4} + 12089766 T^{8} - 5084 p^{4} T^{12} + p^{8} T^{16} \) |
| 47 | \( 1 + 3086 T^{4} + 4580211 T^{8} + 3086 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( 1 - 881 T^{4} - 10532295 T^{8} - 881 p^{4} T^{12} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 162 T^{2} + 12179 T^{4} - 162 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 173 T^{2} + 14289 T^{4} - 173 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 2404 T^{4} - 5755290 T^{8} + 2404 p^{4} T^{12} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 10 T + 146 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( 1 + 16591 T^{4} + 122038521 T^{8} + 16591 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 19 T + 243 T^{2} - 19 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( 1 + 5471 T^{4} + 85671921 T^{8} + 5471 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 333 T^{2} + 43433 T^{4} - 333 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 11239 T^{4} + 183755865 T^{8} + 11239 p^{4} T^{12} + p^{8} T^{16} \) |
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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
−4.05214308377566092779163981436, −3.99804224428995700420988395130, −3.86943465862838171803530231221, −3.86909674073718022070017956713, −3.69230020363874268990680757965, −3.41068936078449281408679539794, −3.37729257873306144054955074759, −3.32768522404168964041972445059, −3.18068389678289552262583963584, −3.02763377765856675825723540275, −2.81327652020980560283863816141, −2.77599928629211002028491996623, −2.74849544593921317877954758797, −2.70065055437919857182417953715, −2.26370423736623481153754182356, −2.09129893194757451022057910227, −1.80313847145750658955185891884, −1.72191252754984689881973519716, −1.38414490653723031343618575146, −1.34240246428388860884425685564, −1.18110071039350545922711705183, −1.04493566107983043255658496641, −0.867601483230896735814702437595, −0.68142534489945786295004895379, −0.47772751160072938499074723736,
0.47772751160072938499074723736, 0.68142534489945786295004895379, 0.867601483230896735814702437595, 1.04493566107983043255658496641, 1.18110071039350545922711705183, 1.34240246428388860884425685564, 1.38414490653723031343618575146, 1.72191252754984689881973519716, 1.80313847145750658955185891884, 2.09129893194757451022057910227, 2.26370423736623481153754182356, 2.70065055437919857182417953715, 2.74849544593921317877954758797, 2.77599928629211002028491996623, 2.81327652020980560283863816141, 3.02763377765856675825723540275, 3.18068389678289552262583963584, 3.32768522404168964041972445059, 3.37729257873306144054955074759, 3.41068936078449281408679539794, 3.69230020363874268990680757965, 3.86909674073718022070017956713, 3.86943465862838171803530231221, 3.99804224428995700420988395130, 4.05214308377566092779163981436
Plot not available for L-functions of degree greater than 10.
