| L(s) = 1 | − 2·2-s + 3-s + 4·4-s + 5·5-s − 2·6-s − 8·8-s − 26·9-s − 10·10-s − 65·11-s + 4·12-s − 13·13-s + 5·15-s + 16·16-s + 73·17-s + 52·18-s + 142·19-s + 20·20-s + 130·22-s + 130·23-s − 8·24-s + 25·25-s + 26·26-s − 53·27-s + 111·29-s − 10·30-s − 256·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.192·3-s + 1/2·4-s + 0.447·5-s − 0.136·6-s − 0.353·8-s − 0.962·9-s − 0.316·10-s − 1.78·11-s + 0.0962·12-s − 0.277·13-s + 0.0860·15-s + 1/4·16-s + 1.04·17-s + 0.680·18-s + 1.71·19-s + 0.223·20-s + 1.25·22-s + 1.17·23-s − 0.0680·24-s + 1/5·25-s + 0.196·26-s − 0.377·27-s + 0.710·29-s − 0.0608·30-s − 1.48·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.314487747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.314487747\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 + 65 T + p^{3} T^{2} \) |
| 13 | \( 1 + p T + p^{3} T^{2} \) |
| 17 | \( 1 - 73 T + p^{3} T^{2} \) |
| 19 | \( 1 - 142 T + p^{3} T^{2} \) |
| 23 | \( 1 - 130 T + p^{3} T^{2} \) |
| 29 | \( 1 - 111 T + p^{3} T^{2} \) |
| 31 | \( 1 + 256 T + p^{3} T^{2} \) |
| 37 | \( 1 + 266 T + p^{3} T^{2} \) |
| 41 | \( 1 - 424 T + p^{3} T^{2} \) |
| 43 | \( 1 - 534 T + p^{3} T^{2} \) |
| 47 | \( 1 - 269 T + p^{3} T^{2} \) |
| 53 | \( 1 + 132 T + p^{3} T^{2} \) |
| 59 | \( 1 - 224 T + p^{3} T^{2} \) |
| 61 | \( 1 - 572 T + p^{3} T^{2} \) |
| 67 | \( 1 + 108 T + p^{3} T^{2} \) |
| 71 | \( 1 - 560 T + p^{3} T^{2} \) |
| 73 | \( 1 + 586 T + p^{3} T^{2} \) |
| 79 | \( 1 - 57 T + p^{3} T^{2} \) |
| 83 | \( 1 + 252 T + p^{3} T^{2} \) |
| 89 | \( 1 - 184 T + p^{3} T^{2} \) |
| 97 | \( 1 - 605 T + p^{3} T^{2} \) |
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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
−10.49918201695087963803867676947, −9.588819382090766394584808389771, −8.856715005425441559867173300201, −7.78476295552921276141224247622, −7.28841588467455711106382076752, −5.65344384420016229735701799243, −5.29151108955600910188879811475, −3.19821706227892982147912468797, −2.45786601116675073852589250564, −0.77297386646065827309668789486,
0.77297386646065827309668789486, 2.45786601116675073852589250564, 3.19821706227892982147912468797, 5.29151108955600910188879811475, 5.65344384420016229735701799243, 7.28841588467455711106382076752, 7.78476295552921276141224247622, 8.856715005425441559867173300201, 9.588819382090766394584808389771, 10.49918201695087963803867676947
