| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2.48·7-s + 8-s + 9-s − 0.963·11-s + 12-s − 5.67·13-s − 2.48·14-s + 16-s − 6.05·17-s + 18-s + 6.48·19-s − 2.48·21-s − 0.963·22-s − 3.53·23-s + 24-s − 5.67·26-s + 27-s − 2.48·28-s − 1.25·29-s + 3.71·31-s + 32-s − 0.963·33-s − 6.05·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.940·7-s + 0.353·8-s + 0.333·9-s − 0.290·11-s + 0.288·12-s − 1.57·13-s − 0.665·14-s + 0.250·16-s − 1.46·17-s + 0.235·18-s + 1.48·19-s − 0.543·21-s − 0.205·22-s − 0.737·23-s + 0.204·24-s − 1.11·26-s + 0.192·27-s − 0.470·28-s − 0.233·29-s + 0.666·31-s + 0.176·32-s − 0.167·33-s − 1.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.48T + 7T^{2} \) |
| 11 | \( 1 + 0.963T + 11T^{2} \) |
| 13 | \( 1 + 5.67T + 13T^{2} \) |
| 17 | \( 1 + 6.05T + 17T^{2} \) |
| 19 | \( 1 - 6.48T + 19T^{2} \) |
| 23 | \( 1 + 3.53T + 23T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 - 3.71T + 31T^{2} \) |
| 37 | \( 1 - 4.65T + 37T^{2} \) |
| 41 | \( 1 + 2.22T + 41T^{2} \) |
| 43 | \( 1 + 9.17T + 43T^{2} \) |
| 47 | \( 1 - 2.57T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 6.61T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 6.38T + 71T^{2} \) |
| 73 | \( 1 + 0.753T + 73T^{2} \) |
| 79 | \( 1 + 3.02T + 79T^{2} \) |
| 83 | \( 1 - 6.86T + 83T^{2} \) |
| 89 | \( 1 + 7.91T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
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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
−7.919691679503064129166835467122, −7.36914239562629561882195666368, −6.65847616401815673294960108437, −5.94629847429992111483265960100, −4.89774889650597722399466553051, −4.40800702286941243653781045551, −3.23554362814402574788384995478, −2.79772144685195123423355367183, −1.82259157794265517084524842757, 0,
1.82259157794265517084524842757, 2.79772144685195123423355367183, 3.23554362814402574788384995478, 4.40800702286941243653781045551, 4.89774889650597722399466553051, 5.94629847429992111483265960100, 6.65847616401815673294960108437, 7.36914239562629561882195666368, 7.919691679503064129166835467122
