| L(s) = 1 | + 2-s + 1.58·3-s + 4-s + 3.51·5-s + 1.58·6-s + 7-s + 8-s − 0.485·9-s + 3.51·10-s + 1.41·11-s + 1.58·12-s + 5.91·13-s + 14-s + 5.57·15-s + 16-s − 3.76·17-s − 0.485·18-s + 0.940·19-s + 3.51·20-s + 1.58·21-s + 1.41·22-s + 1.58·24-s + 7.34·25-s + 5.91·26-s − 5.52·27-s + 28-s − 7.99·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.915·3-s + 0.5·4-s + 1.57·5-s + 0.647·6-s + 0.377·7-s + 0.353·8-s − 0.161·9-s + 1.11·10-s + 0.426·11-s + 0.457·12-s + 1.63·13-s + 0.267·14-s + 1.43·15-s + 0.250·16-s − 0.913·17-s − 0.114·18-s + 0.215·19-s + 0.785·20-s + 0.346·21-s + 0.301·22-s + 0.323·24-s + 1.46·25-s + 1.15·26-s − 1.06·27-s + 0.188·28-s − 1.48·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.986077998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.986077998\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 - 3.51T + 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 + 3.76T + 17T^{2} \) |
| 19 | \( 1 - 0.940T + 19T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 + 7.82T + 31T^{2} \) |
| 37 | \( 1 - 8.76T + 37T^{2} \) |
| 41 | \( 1 - 8.91T + 41T^{2} \) |
| 43 | \( 1 + 2.90T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 6.56T + 53T^{2} \) |
| 59 | \( 1 + 7.32T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 4.95T + 67T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 - 6.22T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 1.43T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−7.88583226859220665567011923653, −7.15484503092856845397836768210, −6.10935952240143283948716001132, −5.97688174583427962993118699319, −5.21992992913341505049443399498, −4.13018299309421198260909191405, −3.59633427534047970042031074220, −2.59945794830323257024452477508, −2.03080233701447579463028328489, −1.28971656103208837846439024361,
1.28971656103208837846439024361, 2.03080233701447579463028328489, 2.59945794830323257024452477508, 3.59633427534047970042031074220, 4.13018299309421198260909191405, 5.21992992913341505049443399498, 5.97688174583427962993118699319, 6.10935952240143283948716001132, 7.15484503092856845397836768210, 7.88583226859220665567011923653
