| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s − 8-s + 9-s + 10-s + 3·11-s + 2·12-s + 13-s − 2·15-s + 16-s + 6·17-s − 18-s + 19-s − 20-s − 3·22-s + 9·23-s − 2·24-s + 25-s − 26-s − 4·27-s + 6·29-s + 2·30-s − 8·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.904·11-s + 0.577·12-s + 0.277·13-s − 0.516·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.639·22-s + 1.87·23-s − 0.408·24-s + 1/5·25-s − 0.196·26-s − 0.769·27-s + 1.11·29-s + 0.365·30-s − 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.495562027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.495562027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.85220388593637973259260583500, −9.798827689345313330649644791210, −9.010457226637545219016876721387, −8.450548827953952381413542526812, −7.54406101859650126734454953475, −6.76369810423149837192636154107, −5.32836056092660803329505468492, −3.70177089518065458380414999112, −2.97795101789227046966578807832, −1.36162716037423381102750436717,
1.36162716037423381102750436717, 2.97795101789227046966578807832, 3.70177089518065458380414999112, 5.32836056092660803329505468492, 6.76369810423149837192636154107, 7.54406101859650126734454953475, 8.450548827953952381413542526812, 9.010457226637545219016876721387, 9.798827689345313330649644791210, 10.85220388593637973259260583500
