| L(s) = 1 | − 2-s + 4-s + 0.561·5-s − 7-s − 8-s − 0.561·10-s − 2.56·11-s + 13-s + 14-s + 16-s − 5.68·17-s + 7.68·19-s + 0.561·20-s + 2.56·22-s + 1.43·23-s − 4.68·25-s − 26-s − 28-s + 5.68·29-s − 10.2·31-s − 32-s + 5.68·34-s − 0.561·35-s − 3.43·37-s − 7.68·38-s − 0.561·40-s − 7.12·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.251·5-s − 0.377·7-s − 0.353·8-s − 0.177·10-s − 0.772·11-s + 0.277·13-s + 0.267·14-s + 0.250·16-s − 1.37·17-s + 1.76·19-s + 0.125·20-s + 0.546·22-s + 0.299·23-s − 0.936·25-s − 0.196·26-s − 0.188·28-s + 1.05·29-s − 1.84·31-s − 0.176·32-s + 0.974·34-s − 0.0949·35-s − 0.565·37-s − 1.24·38-s − 0.0887·40-s − 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 0.561T + 5T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 17 | \( 1 + 5.68T + 17T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 - 1.43T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + 3.43T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 14.2T + 59T^{2} \) |
| 61 | \( 1 + 5.68T + 61T^{2} \) |
| 67 | \( 1 - 1.12T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 0.561T + 73T^{2} \) |
| 79 | \( 1 + 2.87T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.943546469941654705387114337009, −8.368753807190748414751207446163, −7.33181853461224455582846520339, −6.82486306202951059930453674231, −5.75786970631544080979429890362, −5.04332455074417045923424864103, −3.67375410635739582237562205909, −2.72240667229055867481077864761, −1.60297949560182135315821130298, 0,
1.60297949560182135315821130298, 2.72240667229055867481077864761, 3.67375410635739582237562205909, 5.04332455074417045923424864103, 5.75786970631544080979429890362, 6.82486306202951059930453674231, 7.33181853461224455582846520339, 8.368753807190748414751207446163, 8.943546469941654705387114337009
