| L(s) = 1 | + 1.90·2-s + 1.62·4-s − 5-s + 4.42·7-s − 0.719·8-s − 1.90·10-s + 0.622·13-s + 8.42·14-s − 4.61·16-s − 5.18·17-s − 7.05·19-s − 1.62·20-s − 8.85·23-s + 25-s + 1.18·26-s + 7.18·28-s − 7.80·29-s + 2.75·31-s − 7.34·32-s − 9.86·34-s − 4.42·35-s − 2·37-s − 13.4·38-s + 0.719·40-s − 0.193·41-s − 5.67·43-s − 16.8·46-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.811·4-s − 0.447·5-s + 1.67·7-s − 0.254·8-s − 0.601·10-s + 0.172·13-s + 2.25·14-s − 1.15·16-s − 1.25·17-s − 1.61·19-s − 0.362·20-s − 1.84·23-s + 0.200·25-s + 0.232·26-s + 1.35·28-s − 1.44·29-s + 0.494·31-s − 1.29·32-s − 1.69·34-s − 0.748·35-s − 0.328·37-s − 2.17·38-s + 0.113·40-s − 0.0302·41-s − 0.865·43-s − 2.48·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 13 | \( 1 - 0.622T + 13T^{2} \) |
| 17 | \( 1 + 5.18T + 17T^{2} \) |
| 19 | \( 1 + 7.05T + 19T^{2} \) |
| 23 | \( 1 + 8.85T + 23T^{2} \) |
| 29 | \( 1 + 7.80T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 0.193T + 41T^{2} \) |
| 43 | \( 1 + 5.67T + 43T^{2} \) |
| 47 | \( 1 - 2.75T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 4.85T + 59T^{2} \) |
| 61 | \( 1 + 6.85T + 61T^{2} \) |
| 67 | \( 1 + 1.24T + 67T^{2} \) |
| 71 | \( 1 + 2.75T + 71T^{2} \) |
| 73 | \( 1 + 4.23T + 73T^{2} \) |
| 79 | \( 1 + 8.56T + 79T^{2} \) |
| 83 | \( 1 - 0.133T + 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 - 7.24T + 97T^{2} \) |
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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
−7.74147011178756666919746107497, −6.96132561806301006619856691523, −6.12047542130501290273707517048, −5.53584458108554401907421425031, −4.58253487885474746207307838843, −4.32357601484572013268857337832, −3.67472587995782495781316632966, −2.33071398447234227028089519905, −1.85343221620870638175535135244, 0,
1.85343221620870638175535135244, 2.33071398447234227028089519905, 3.67472587995782495781316632966, 4.32357601484572013268857337832, 4.58253487885474746207307838843, 5.53584458108554401907421425031, 6.12047542130501290273707517048, 6.96132561806301006619856691523, 7.74147011178756666919746107497
