| L(s) = 1 | − 0.959·3-s − 1.46·5-s − 0.668·7-s − 2.07·9-s + 4.64·11-s + 5.51·13-s + 1.40·15-s + 3.78·17-s − 19-s + 0.641·21-s + 1.56·23-s − 2.84·25-s + 4.87·27-s − 3.75·29-s − 1.53·31-s − 4.45·33-s + 0.980·35-s + 8.35·37-s − 5.29·39-s − 7.65·41-s + 6.96·43-s + 3.04·45-s − 1.27·47-s − 6.55·49-s − 3.63·51-s + 12.5·53-s − 6.81·55-s + ⋯ |
| L(s) = 1 | − 0.554·3-s − 0.655·5-s − 0.252·7-s − 0.693·9-s + 1.40·11-s + 1.52·13-s + 0.363·15-s + 0.918·17-s − 0.229·19-s + 0.140·21-s + 0.325·23-s − 0.569·25-s + 0.938·27-s − 0.697·29-s − 0.275·31-s − 0.776·33-s + 0.165·35-s + 1.37·37-s − 0.847·39-s − 1.19·41-s + 1.06·43-s + 0.454·45-s − 0.185·47-s − 0.936·49-s − 0.509·51-s + 1.72·53-s − 0.919·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.404374969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.404374969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| good | 3 | \( 1 + 0.959T + 3T^{2} \) |
| 5 | \( 1 + 1.46T + 5T^{2} \) |
| 7 | \( 1 + 0.668T + 7T^{2} \) |
| 11 | \( 1 - 4.64T + 11T^{2} \) |
| 13 | \( 1 - 5.51T + 13T^{2} \) |
| 17 | \( 1 - 3.78T + 17T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 + 3.75T + 29T^{2} \) |
| 31 | \( 1 + 1.53T + 31T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 - 6.96T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 4.99T + 59T^{2} \) |
| 61 | \( 1 + 7.98T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 8.23T + 73T^{2} \) |
| 83 | \( 1 - 1.90T + 83T^{2} \) |
| 89 | \( 1 - 8.65T + 89T^{2} \) |
| 97 | \( 1 - 2.10T + 97T^{2} \) |
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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
−8.095748345900177176140377541735, −7.37114618312891326288159904229, −6.44241222473471990120996536261, −6.06412055129443255221217130930, −5.38475025451378294609680783743, −4.26102096033463116022659098932, −3.71940168277006036216141467700, −3.03869995093605285629568412269, −1.59574119224155456481672060884, −0.67754033970125683458828525120,
0.67754033970125683458828525120, 1.59574119224155456481672060884, 3.03869995093605285629568412269, 3.71940168277006036216141467700, 4.26102096033463116022659098932, 5.38475025451378294609680783743, 6.06412055129443255221217130930, 6.44241222473471990120996536261, 7.37114618312891326288159904229, 8.095748345900177176140377541735
