| L(s) = 1 | − 2-s + 4-s − 0.532·5-s − 2.87·7-s − 8-s + 0.532·10-s + 1.18·11-s − 0.773·13-s + 2.87·14-s + 16-s + 6.59·17-s − 1.34·19-s − 0.532·20-s − 1.18·22-s − 2.94·23-s − 4.71·25-s + 0.773·26-s − 2.87·28-s + 4.26·29-s + 4.17·31-s − 32-s − 6.59·34-s + 1.53·35-s − 9.02·37-s + 1.34·38-s + 0.532·40-s − 5.17·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.237·5-s − 1.08·7-s − 0.353·8-s + 0.168·10-s + 0.357·11-s − 0.214·13-s + 0.769·14-s + 0.250·16-s + 1.59·17-s − 0.309·19-s − 0.118·20-s − 0.252·22-s − 0.613·23-s − 0.943·25-s + 0.151·26-s − 0.544·28-s + 0.792·29-s + 0.748·31-s − 0.176·32-s − 1.13·34-s + 0.258·35-s − 1.48·37-s + 0.218·38-s + 0.0841·40-s − 0.807·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 + 0.532T + 5T^{2} \) |
| 7 | \( 1 + 2.87T + 7T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 13 | \( 1 + 0.773T + 13T^{2} \) |
| 17 | \( 1 - 6.59T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 + 2.94T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 - 4.17T + 31T^{2} \) |
| 37 | \( 1 + 9.02T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 - 1.04T + 43T^{2} \) |
| 47 | \( 1 - 8.61T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 + 7.29T + 59T^{2} \) |
| 61 | \( 1 + 0.0564T + 61T^{2} \) |
| 67 | \( 1 - 9.43T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 9.29T + 73T^{2} \) |
| 79 | \( 1 + 2.57T + 79T^{2} \) |
| 83 | \( 1 - 7.84T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.110101820525518784818416557364, −7.42362669568095557824902335656, −6.69950395528874181409006074460, −6.04266341002049480781776292703, −5.27561896728964916097160413630, −4.00674630171156611346783483968, −3.36150843786468517880454984592, −2.44982007721010542055685786850, −1.21694550193589941713477804393, 0,
1.21694550193589941713477804393, 2.44982007721010542055685786850, 3.36150843786468517880454984592, 4.00674630171156611346783483968, 5.27561896728964916097160413630, 6.04266341002049480781776292703, 6.69950395528874181409006074460, 7.42362669568095557824902335656, 8.110101820525518784818416557364
