| L(s) = 1 | + 2.12·2-s + 2.51·4-s − 3.64·7-s + 1.09·8-s − 11-s − 1.51·13-s − 7.73·14-s − 2.70·16-s + 1.15·17-s + 2.60·19-s − 2.12·22-s − 5.73·23-s − 3.21·26-s − 9.15·28-s − 6.24·29-s + 5.51·31-s − 7.93·32-s + 2.45·34-s − 0.454·37-s + 5.54·38-s − 4.12·41-s − 11.7·43-s − 2.51·44-s − 12.1·46-s − 3.48·47-s + 6.24·49-s − 3.81·52-s + ⋯ |
| L(s) = 1 | + 1.50·2-s + 1.25·4-s − 1.37·7-s + 0.387·8-s − 0.301·11-s − 0.420·13-s − 2.06·14-s − 0.676·16-s + 0.280·17-s + 0.598·19-s − 0.453·22-s − 1.19·23-s − 0.631·26-s − 1.73·28-s − 1.16·29-s + 0.990·31-s − 1.40·32-s + 0.420·34-s − 0.0747·37-s + 0.899·38-s − 0.644·41-s − 1.78·43-s − 0.379·44-s − 1.79·46-s − 0.508·47-s + 0.892·49-s − 0.528·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 13 | \( 1 + 1.51T + 13T^{2} \) |
| 17 | \( 1 - 1.15T + 17T^{2} \) |
| 19 | \( 1 - 2.60T + 19T^{2} \) |
| 23 | \( 1 + 5.73T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 5.51T + 31T^{2} \) |
| 37 | \( 1 + 0.454T + 37T^{2} \) |
| 41 | \( 1 + 4.12T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 7.73T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 8.51T + 71T^{2} \) |
| 73 | \( 1 - 9.21T + 73T^{2} \) |
| 79 | \( 1 - 5.09T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 6.77T + 97T^{2} \) |
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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
−8.458737548377580695036661566539, −7.50038649977883579061731863671, −6.64789392437869995997390514080, −6.12720421403050291464411139296, −5.34610289527388845902917907609, −4.58429675327292667735094426449, −3.52002541649683301792237737810, −3.16049985051916385605617533010, −2.05164760275002550134181489637, 0,
2.05164760275002550134181489637, 3.16049985051916385605617533010, 3.52002541649683301792237737810, 4.58429675327292667735094426449, 5.34610289527388845902917907609, 6.12720421403050291464411139296, 6.64789392437869995997390514080, 7.50038649977883579061731863671, 8.458737548377580695036661566539
