L(s) = 1 | + 2-s + 1.69·3-s + 4-s + 4.29·5-s + 1.69·6-s + 5.04·7-s + 8-s − 0.124·9-s + 4.29·10-s − 4.35·11-s + 1.69·12-s − 0.211·13-s + 5.04·14-s + 7.27·15-s + 16-s − 1.15·17-s − 0.124·18-s + 19-s + 4.29·20-s + 8.55·21-s − 4.35·22-s + 3.33·23-s + 1.69·24-s + 13.4·25-s − 0.211·26-s − 5.29·27-s + 5.04·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.979·3-s + 0.5·4-s + 1.91·5-s + 0.692·6-s + 1.90·7-s + 0.353·8-s − 0.0414·9-s + 1.35·10-s − 1.31·11-s + 0.489·12-s − 0.0587·13-s + 1.34·14-s + 1.87·15-s + 0.250·16-s − 0.280·17-s − 0.0292·18-s + 0.229·19-s + 0.959·20-s + 1.86·21-s − 0.928·22-s + 0.694·23-s + 0.346·24-s + 2.68·25-s − 0.0415·26-s − 1.01·27-s + 0.952·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.064718595\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.064718595\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 1.69T + 3T^{2} \) |
| 5 | \( 1 - 4.29T + 5T^{2} \) |
| 7 | \( 1 - 5.04T + 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 + 0.211T + 13T^{2} \) |
| 17 | \( 1 + 1.15T + 17T^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 - 8.92T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 0.663T + 41T^{2} \) |
| 43 | \( 1 - 0.0899T + 43T^{2} \) |
| 47 | \( 1 + 6.57T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 + 0.292T + 59T^{2} \) |
| 61 | \( 1 - 8.07T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 - 9.34T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 - 7.25T + 97T^{2} \) |
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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
−7.88175488865283481405538423901, −7.19168065286424656500648157760, −6.29251868903638494078151285962, −5.47747705100380603881492455297, −5.06902050055127378011068229293, −4.62046871822339076698807765062, −3.19959216225661258393420864016, −2.60779826478076792977594353602, −1.97735761645994636538086895332, −1.41528195896831293775857965193,
1.41528195896831293775857965193, 1.97735761645994636538086895332, 2.60779826478076792977594353602, 3.19959216225661258393420864016, 4.62046871822339076698807765062, 5.06902050055127378011068229293, 5.47747705100380603881492455297, 6.29251868903638494078151285962, 7.19168065286424656500648157760, 7.88175488865283481405538423901