| L(s) = 1 | + 2·2-s − 2·3-s + 4-s + 2·5-s − 4·6-s − 2·8-s + 9-s + 4·10-s − 12·11-s − 2·12-s + 8·13-s − 4·15-s − 4·16-s + 2·17-s + 2·18-s + 2·20-s − 24·22-s + 6·23-s + 4·24-s + 25-s + 16·26-s + 2·27-s − 10·29-s − 8·30-s − 8·31-s − 2·32-s + 24·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s − 1.63·6-s − 0.707·8-s + 1/3·9-s + 1.26·10-s − 3.61·11-s − 0.577·12-s + 2.21·13-s − 1.03·15-s − 16-s + 0.485·17-s + 0.471·18-s + 0.447·20-s − 5.11·22-s + 1.25·23-s + 0.816·24-s + 1/5·25-s + 3.13·26-s + 0.384·27-s − 1.85·29-s − 1.46·30-s − 1.43·31-s − 0.353·32-s + 4.17·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1181515147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1181515147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 5 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 2 T - 12 T^{2} + 36 T^{3} - 101 T^{4} + 36 p T^{5} - 12 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 10 T + 36 T^{2} + 60 T^{3} + 355 T^{4} + 60 p T^{5} + 36 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4 T - 51 T^{2} + 60 T^{3} + 2152 T^{4} + 60 p T^{5} - 51 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 65 T^{2} + 1416 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 8 T + 6 T^{2} - 480 T^{3} - 3893 T^{4} - 480 p T^{5} + 6 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T - 100 T^{2} - 36 T^{3} + 6851 T^{4} - 36 p T^{5} - 100 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T + 32 T^{2} + 420 T^{3} + 9275 T^{4} + 420 p T^{5} + 32 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T + 56 T^{2} - 972 T^{3} - 7197 T^{4} - 972 p T^{5} + 56 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 18 T + 116 T^{2} + 1116 T^{3} + 14283 T^{4} + 1116 p T^{5} + 116 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8 T - 34 T^{2} - 480 T^{3} - 1693 T^{4} - 480 p T^{5} - 34 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8 T - 111 T^{2} + 24 T^{3} + 17008 T^{4} + 24 p T^{5} - 111 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2 T - 20 T^{2} + 340 T^{3} - 9221 T^{4} + 340 p T^{5} - 20 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.48364351778155417051606713270, −7.38241732797961059606230005261, −7.29974219334851439077275392342, −7.01502423702084290258210777783, −6.84818255786111597811264061326, −6.08073476862071474196335228095, −6.07795267019598013796138337880, −5.90690892526157251968392702098, −5.67232768690701720642479908622, −5.51709216339812757143367484910, −5.40540472116662672330317183893, −5.15533378634657451355223621147, −5.07083348932359159497437956352, −4.53924983221258163860377843305, −4.18111194001153863925359685124, −4.12981803448696506781788778313, −3.53901899644976064063881312067, −3.49853706084626224085784772319, −3.05096724082480074397578310433, −2.65727370149076214949415935096, −2.62528540222150442095753642479, −2.08740078002895903077963268781, −1.50848181995704059546899984299, −1.23427886987856013288839635268, −0.089032823645111286563519576607,
0.089032823645111286563519576607, 1.23427886987856013288839635268, 1.50848181995704059546899984299, 2.08740078002895903077963268781, 2.62528540222150442095753642479, 2.65727370149076214949415935096, 3.05096724082480074397578310433, 3.49853706084626224085784772319, 3.53901899644976064063881312067, 4.12981803448696506781788778313, 4.18111194001153863925359685124, 4.53924983221258163860377843305, 5.07083348932359159497437956352, 5.15533378634657451355223621147, 5.40540472116662672330317183893, 5.51709216339812757143367484910, 5.67232768690701720642479908622, 5.90690892526157251968392702098, 6.07795267019598013796138337880, 6.08073476862071474196335228095, 6.84818255786111597811264061326, 7.01502423702084290258210777783, 7.29974219334851439077275392342, 7.38241732797961059606230005261, 7.48364351778155417051606713270
