| L(s) = 1 | − 2.43·2-s + 2.36·3-s + 3.92·4-s + 5-s − 5.75·6-s − 7-s − 4.68·8-s + 2.59·9-s − 2.43·10-s + 9.28·12-s − 1.34·13-s + 2.43·14-s + 2.36·15-s + 3.56·16-s − 1.53·17-s − 6.31·18-s + 1.50·19-s + 3.92·20-s − 2.36·21-s + 2.62·23-s − 11.0·24-s + 25-s + 3.26·26-s − 0.958·27-s − 3.92·28-s − 7.31·29-s − 5.75·30-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 1.36·3-s + 1.96·4-s + 0.447·5-s − 2.35·6-s − 0.377·7-s − 1.65·8-s + 0.864·9-s − 0.769·10-s + 2.68·12-s − 0.371·13-s + 0.650·14-s + 0.610·15-s + 0.890·16-s − 0.371·17-s − 1.48·18-s + 0.344·19-s + 0.877·20-s − 0.516·21-s + 0.548·23-s − 2.26·24-s + 0.200·25-s + 0.639·26-s − 0.184·27-s − 0.741·28-s − 1.35·29-s − 1.05·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 3 | \( 1 - 2.36T + 3T^{2} \) |
| 13 | \( 1 + 1.34T + 13T^{2} \) |
| 17 | \( 1 + 1.53T + 17T^{2} \) |
| 19 | \( 1 - 1.50T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 + 7.31T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 4.01T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 + 6.51T + 47T^{2} \) |
| 53 | \( 1 + 0.974T + 53T^{2} \) |
| 59 | \( 1 + 0.145T + 59T^{2} \) |
| 61 | \( 1 + 4.83T + 61T^{2} \) |
| 67 | \( 1 + 6.74T + 67T^{2} \) |
| 71 | \( 1 - 4.55T + 71T^{2} \) |
| 73 | \( 1 - 5.13T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.273217166449953938804627257170, −7.35401244485502855990033885333, −7.22296809738086561006719783875, −6.17391677770821248876424645151, −5.18181753617336024621698011764, −3.79887494849885598640095135593, −2.99503999804093788544092266048, −2.17929196518869179665182134975, −1.53625436142024455365372786184, 0,
1.53625436142024455365372786184, 2.17929196518869179665182134975, 2.99503999804093788544092266048, 3.79887494849885598640095135593, 5.18181753617336024621698011764, 6.17391677770821248876424645151, 7.22296809738086561006719783875, 7.35401244485502855990033885333, 8.273217166449953938804627257170
