| L(s) = 1 | − 1.63·2-s − 2.57·3-s + 0.687·4-s − 4.16·5-s + 4.21·6-s − 3.92·7-s + 2.15·8-s + 3.61·9-s + 6.82·10-s − 1.76·12-s + 2.96·13-s + 6.42·14-s + 10.7·15-s − 4.90·16-s − 2.15·17-s − 5.92·18-s − 0.0888·19-s − 2.86·20-s + 10.0·21-s − 3.24·23-s − 5.53·24-s + 12.3·25-s − 4.86·26-s − 1.58·27-s − 2.69·28-s + 7.68·29-s − 17.5·30-s + ⋯ |
| L(s) = 1 | − 1.15·2-s − 1.48·3-s + 0.343·4-s − 1.86·5-s + 1.72·6-s − 1.48·7-s + 0.760·8-s + 1.20·9-s + 2.15·10-s − 0.510·12-s + 0.823·13-s + 1.71·14-s + 2.76·15-s − 1.22·16-s − 0.523·17-s − 1.39·18-s − 0.0203·19-s − 0.639·20-s + 2.20·21-s − 0.676·23-s − 1.12·24-s + 2.46·25-s − 0.954·26-s − 0.304·27-s − 0.509·28-s + 1.42·29-s − 3.20·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3034799887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3034799887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.63T + 2T^{2} \) |
| 3 | \( 1 + 2.57T + 3T^{2} \) |
| 5 | \( 1 + 4.16T + 5T^{2} \) |
| 7 | \( 1 + 3.92T + 7T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 + 2.15T + 17T^{2} \) |
| 19 | \( 1 + 0.0888T + 19T^{2} \) |
| 23 | \( 1 + 3.24T + 23T^{2} \) |
| 29 | \( 1 - 7.68T + 29T^{2} \) |
| 31 | \( 1 - 5.63T + 31T^{2} \) |
| 37 | \( 1 - 6.87T + 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 - 9.62T + 47T^{2} \) |
| 53 | \( 1 - 0.282T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 67 | \( 1 + 4.33T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 6.84T + 73T^{2} \) |
| 79 | \( 1 - 9.16T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 1.30T + 89T^{2} \) |
| 97 | \( 1 - 9.52T + 97T^{2} \) |
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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
−7.890835353489606344629997992725, −7.27643204644909010493329553058, −6.46560086838289451044499197466, −6.27975530135247766526753836552, −5.01692583284227622017734760550, −4.26475910111417603102918906849, −3.79777310274084259222498474511, −2.67960861919389017721144483726, −0.76165458880534939208016511356, −0.60959864361180724736572978067,
0.60959864361180724736572978067, 0.76165458880534939208016511356, 2.67960861919389017721144483726, 3.79777310274084259222498474511, 4.26475910111417603102918906849, 5.01692583284227622017734760550, 6.27975530135247766526753836552, 6.46560086838289451044499197466, 7.27643204644909010493329553058, 7.890835353489606344629997992725
