| L(s) = 1 | + 1.53·2-s − 3-s + 0.360·4-s − 1.53·6-s + 1.49·7-s − 2.51·8-s + 9-s − 2.35·11-s − 0.360·12-s + 1.34·13-s + 2.29·14-s − 4.59·16-s − 2.19·17-s + 1.53·18-s + 5.71·19-s − 1.49·21-s − 3.62·22-s + 8.79·23-s + 2.51·24-s + 2.07·26-s − 27-s + 0.539·28-s + 7.90·29-s − 3.69·31-s − 2.01·32-s + 2.35·33-s − 3.37·34-s + ⋯ |
| L(s) = 1 | + 1.08·2-s − 0.577·3-s + 0.180·4-s − 0.627·6-s + 0.565·7-s − 0.890·8-s + 0.333·9-s − 0.710·11-s − 0.104·12-s + 0.374·13-s + 0.614·14-s − 1.14·16-s − 0.532·17-s + 0.362·18-s + 1.31·19-s − 0.326·21-s − 0.771·22-s + 1.83·23-s + 0.514·24-s + 0.406·26-s − 0.192·27-s + 0.102·28-s + 1.46·29-s − 0.664·31-s − 0.356·32-s + 0.410·33-s − 0.578·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.326712307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.326712307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 7 | \( 1 - 1.49T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 1.34T + 13T^{2} \) |
| 17 | \( 1 + 2.19T + 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 - 8.79T + 23T^{2} \) |
| 29 | \( 1 - 7.90T + 29T^{2} \) |
| 31 | \( 1 + 3.69T + 31T^{2} \) |
| 37 | \( 1 - 9.75T + 37T^{2} \) |
| 41 | \( 1 - 1.85T + 41T^{2} \) |
| 43 | \( 1 + 8.01T + 43T^{2} \) |
| 47 | \( 1 - 6.66T + 47T^{2} \) |
| 53 | \( 1 - 4.17T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 4.31T + 67T^{2} \) |
| 71 | \( 1 - 5.77T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 0.224T + 83T^{2} \) |
| 89 | \( 1 - 0.429T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.215617956356189808632191975704, −8.449018637214262268297213408893, −7.45851889562827019253786728376, −6.63167739561058230575024776923, −5.75985625845760658252803813626, −5.02745993086781062273325002164, −4.61773042029849029755732659563, −3.46697208012111020525918845717, −2.57875479947324953413614594800, −0.931835801920470098868480984877,
0.931835801920470098868480984877, 2.57875479947324953413614594800, 3.46697208012111020525918845717, 4.61773042029849029755732659563, 5.02745993086781062273325002164, 5.75985625845760658252803813626, 6.63167739561058230575024776923, 7.45851889562827019253786728376, 8.449018637214262268297213408893, 9.215617956356189808632191975704
