| L(s) = 1 | + 2-s − 2.84·3-s + 4-s − 0.961·5-s − 2.84·6-s + 8-s + 5.08·9-s − 0.961·10-s − 0.868·11-s − 2.84·12-s − 2.34·13-s + 2.73·15-s + 16-s + 6.29·17-s + 5.08·18-s − 0.246·19-s − 0.961·20-s − 0.868·22-s − 8.33·23-s − 2.84·24-s − 4.07·25-s − 2.34·26-s − 5.92·27-s − 7.72·29-s + 2.73·30-s + 6.88·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.64·3-s + 0.5·4-s − 0.430·5-s − 1.16·6-s + 0.353·8-s + 1.69·9-s − 0.304·10-s − 0.261·11-s − 0.820·12-s − 0.650·13-s + 0.705·15-s + 0.250·16-s + 1.52·17-s + 1.19·18-s − 0.0565·19-s − 0.215·20-s − 0.185·22-s − 1.73·23-s − 0.580·24-s − 0.815·25-s − 0.460·26-s − 1.13·27-s − 1.43·29-s + 0.499·30-s + 1.23·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7742 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.165588432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.165588432\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 2.84T + 3T^{2} \) |
| 5 | \( 1 + 0.961T + 5T^{2} \) |
| 11 | \( 1 + 0.868T + 11T^{2} \) |
| 13 | \( 1 + 2.34T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 19 | \( 1 + 0.246T + 19T^{2} \) |
| 23 | \( 1 + 8.33T + 23T^{2} \) |
| 29 | \( 1 + 7.72T + 29T^{2} \) |
| 31 | \( 1 - 6.88T + 31T^{2} \) |
| 37 | \( 1 - 9.75T + 37T^{2} \) |
| 41 | \( 1 + 3.68T + 41T^{2} \) |
| 43 | \( 1 - 6.04T + 43T^{2} \) |
| 47 | \( 1 + 5.07T + 47T^{2} \) |
| 53 | \( 1 + 1.58T + 53T^{2} \) |
| 59 | \( 1 - 7.84T + 59T^{2} \) |
| 61 | \( 1 - 0.474T + 61T^{2} \) |
| 67 | \( 1 - 8.63T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 83 | \( 1 - 3.20T + 83T^{2} \) |
| 89 | \( 1 + 18.7T + 89T^{2} \) |
| 97 | \( 1 - 6.64T + 97T^{2} \) |
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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
−7.87953397376806979277501008893, −6.94085496934084382032323619652, −6.26903209497727975563367890356, −5.56500802839931760585164851571, −5.35432447309502138294560690311, −4.29539245531200623934052277787, −3.94518977470868005907686678106, −2.77932173984426719598504731703, −1.67481190000925759092941681700, −0.52725683372729625202399545432,
0.52725683372729625202399545432, 1.67481190000925759092941681700, 2.77932173984426719598504731703, 3.94518977470868005907686678106, 4.29539245531200623934052277787, 5.35432447309502138294560690311, 5.56500802839931760585164851571, 6.26903209497727975563367890356, 6.94085496934084382032323619652, 7.87953397376806979277501008893
