| L(s) = 1 | − 2.09·2-s + 2.40·4-s − 2.53·5-s − 7-s − 0.839·8-s + 5.31·10-s − 5.69·13-s + 2.09·14-s − 3.03·16-s + 0.996·17-s − 6.02·19-s − 6.07·20-s − 1.25·23-s + 1.41·25-s + 11.9·26-s − 2.40·28-s + 1.33·29-s − 1.85·31-s + 8.05·32-s − 2.09·34-s + 2.53·35-s + 9.65·37-s + 12.6·38-s + 2.12·40-s + 10.0·41-s + 5.04·43-s + 2.63·46-s + ⋯ |
| L(s) = 1 | − 1.48·2-s + 1.20·4-s − 1.13·5-s − 0.377·7-s − 0.296·8-s + 1.67·10-s − 1.58·13-s + 0.560·14-s − 0.759·16-s + 0.241·17-s − 1.38·19-s − 1.35·20-s − 0.262·23-s + 0.282·25-s + 2.34·26-s − 0.453·28-s + 0.248·29-s − 0.332·31-s + 1.42·32-s − 0.358·34-s + 0.428·35-s + 1.58·37-s + 2.05·38-s + 0.336·40-s + 1.56·41-s + 0.769·43-s + 0.389·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.09T + 2T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 13 | \( 1 + 5.69T + 13T^{2} \) |
| 17 | \( 1 - 0.996T + 17T^{2} \) |
| 19 | \( 1 + 6.02T + 19T^{2} \) |
| 23 | \( 1 + 1.25T + 23T^{2} \) |
| 29 | \( 1 - 1.33T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 - 9.65T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 - 5.04T + 43T^{2} \) |
| 47 | \( 1 + 1.03T + 47T^{2} \) |
| 53 | \( 1 - 7.43T + 53T^{2} \) |
| 59 | \( 1 - 6.55T + 59T^{2} \) |
| 61 | \( 1 + 3.76T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 6.41T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 + 2.05T + 97T^{2} \) |
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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
−7.61138547355424386550824222335, −7.22628489956434042228455687349, −6.51525223686964604359954852093, −5.56476365649448013889026033459, −4.39804208691638147655634024365, −4.10105935201871676108555858107, −2.75666672746211419244918912599, −2.17712480792533747252314886275, −0.791772636095629880227749112336, 0,
0.791772636095629880227749112336, 2.17712480792533747252314886275, 2.75666672746211419244918912599, 4.10105935201871676108555858107, 4.39804208691638147655634024365, 5.56476365649448013889026033459, 6.51525223686964604359954852093, 7.22628489956434042228455687349, 7.61138547355424386550824222335
