| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 3·9-s − 10-s − 11-s − 6·13-s − 14-s + 16-s − 2·17-s + 3·18-s − 4·19-s + 20-s + 22-s − 4·23-s + 25-s + 6·26-s + 28-s + 6·29-s − 32-s + 2·34-s + 35-s − 3·36-s − 2·37-s + 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 9-s − 0.316·10-s − 0.301·11-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.917·19-s + 0.223·20-s + 0.213·22-s − 0.834·23-s + 1/5·25-s + 1.17·26-s + 0.188·28-s + 1.11·29-s − 0.176·32-s + 0.342·34-s + 0.169·35-s − 1/2·36-s − 0.328·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−10.04063769026327802642478703475, −8.916730340641706279929786137233, −8.363523700282027049152069102287, −7.41245154202477197034737063101, −6.48198458599832188346302162345, −5.51130120255433556946095894516, −4.55174504902523336463882421353, −2.84631505847919077884555890958, −2.02488564421173045830172664749, 0,
2.02488564421173045830172664749, 2.84631505847919077884555890958, 4.55174504902523336463882421353, 5.51130120255433556946095894516, 6.48198458599832188346302162345, 7.41245154202477197034737063101, 8.363523700282027049152069102287, 8.916730340641706279929786137233, 10.04063769026327802642478703475
