L(s) = 1 | − 2-s + 0.381·3-s − 4-s + 1.61·5-s − 0.381·6-s − 7-s + 3·8-s − 2.85·9-s − 1.61·10-s − 0.381·11-s − 0.381·12-s + 3.38·13-s + 14-s + 0.618·15-s − 16-s + 1.23·17-s + 2.85·18-s + 7.09·19-s − 1.61·20-s − 0.381·21-s + 0.381·22-s − 6.09·23-s + 1.14·24-s − 2.38·25-s − 3.38·26-s − 2.23·27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.220·3-s − 0.5·4-s + 0.723·5-s − 0.155·6-s − 0.377·7-s + 1.06·8-s − 0.951·9-s − 0.511·10-s − 0.115·11-s − 0.110·12-s + 0.937·13-s + 0.267·14-s + 0.159·15-s − 0.250·16-s + 0.299·17-s + 0.672·18-s + 1.62·19-s − 0.361·20-s − 0.0833·21-s + 0.0814·22-s − 1.26·23-s + 0.233·24-s − 0.476·25-s − 0.663·26-s − 0.430·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 863 | \( 1 + T \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 11 | \( 1 + 0.381T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 23 | \( 1 + 6.09T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 8.85T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 - 7.70T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 8.38T + 59T^{2} \) |
| 61 | \( 1 - 0.291T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 + 11.3T + 79T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 1.85T + 97T^{2} \) |
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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
−7.903593883955323280920483257250, −7.24082249208048815642605076363, −6.19755158783371892609049136812, −5.61967847425132467550273266263, −5.04506575913172768637878753450, −3.80619497765751653309732269049, −3.29277313806043495190102038987, −2.12493957335690480931053399602, −1.22875976686850573077630225103, 0,
1.22875976686850573077630225103, 2.12493957335690480931053399602, 3.29277313806043495190102038987, 3.80619497765751653309732269049, 5.04506575913172768637878753450, 5.61967847425132467550273266263, 6.19755158783371892609049136812, 7.24082249208048815642605076363, 7.903593883955323280920483257250