| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 2·6-s + 3·7-s + 9-s + 2·11-s + 2·12-s − 13-s − 6·14-s − 4·16-s − 2·17-s − 2·18-s − 5·19-s + 3·21-s − 4·22-s − 6·23-s + 2·26-s + 27-s + 6·28-s + 10·29-s − 3·31-s + 8·32-s + 2·33-s + 4·34-s + 2·36-s − 2·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.577·12-s − 0.277·13-s − 1.60·14-s − 16-s − 0.485·17-s − 0.471·18-s − 1.14·19-s + 0.654·21-s − 0.852·22-s − 1.25·23-s + 0.392·26-s + 0.192·27-s + 1.13·28-s + 1.85·29-s − 0.538·31-s + 1.41·32-s + 0.348·33-s + 0.685·34-s + 1/3·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6272349294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6272349294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.63360099431736791767125137970, −13.71100770465530571761239213204, −12.06535678251561534347450688093, −10.88974094536361104567387978062, −9.903619965777680376182050580499, −8.655074250265874594654667643843, −8.114757177440921419069207299350, −6.77287251257420576381216593643, −4.48018036005395779311676580836, −1.90319955654708104595670502735,
1.90319955654708104595670502735, 4.48018036005395779311676580836, 6.77287251257420576381216593643, 8.114757177440921419069207299350, 8.655074250265874594654667643843, 9.903619965777680376182050580499, 10.88974094536361104567387978062, 12.06535678251561534347450688093, 13.71100770465530571761239213204, 14.63360099431736791767125137970
