L(s) = 1 | − 2.58·2-s − 2.87·3-s + 4.68·4-s − 1.51·5-s + 7.44·6-s + 0.161·7-s − 6.93·8-s + 5.28·9-s + 3.91·10-s + 11-s − 13.4·12-s − 6.99·13-s − 0.417·14-s + 4.36·15-s + 8.56·16-s − 4.66·17-s − 13.6·18-s + 4.03·19-s − 7.09·20-s − 0.464·21-s − 2.58·22-s + 4.97·23-s + 19.9·24-s − 2.70·25-s + 18.0·26-s − 6.58·27-s + 0.756·28-s + ⋯ |
L(s) = 1 | − 1.82·2-s − 1.66·3-s + 2.34·4-s − 0.677·5-s + 3.03·6-s + 0.0610·7-s − 2.45·8-s + 1.76·9-s + 1.23·10-s + 0.301·11-s − 3.89·12-s − 1.94·13-s − 0.111·14-s + 1.12·15-s + 2.14·16-s − 1.13·17-s − 3.22·18-s + 0.926·19-s − 1.58·20-s − 0.101·21-s − 0.551·22-s + 1.03·23-s + 4.07·24-s − 0.540·25-s + 3.54·26-s − 1.26·27-s + 0.142·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 547 | \( 1 - T \) |
good | 2 | \( 1 + 2.58T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 5 | \( 1 + 1.51T + 5T^{2} \) |
| 7 | \( 1 - 0.161T + 7T^{2} \) |
| 13 | \( 1 + 6.99T + 13T^{2} \) |
| 17 | \( 1 + 4.66T + 17T^{2} \) |
| 19 | \( 1 - 4.03T + 19T^{2} \) |
| 23 | \( 1 - 4.97T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 31 | \( 1 + 0.546T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 - 4.07T + 41T^{2} \) |
| 43 | \( 1 - 5.47T + 43T^{2} \) |
| 47 | \( 1 + 2.72T + 47T^{2} \) |
| 53 | \( 1 - 13.7T + 53T^{2} \) |
| 59 | \( 1 + 0.114T + 59T^{2} \) |
| 61 | \( 1 + 2.36T + 61T^{2} \) |
| 67 | \( 1 - 4.13T + 67T^{2} \) |
| 71 | \( 1 - 5.67T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 7.56T + 79T^{2} \) |
| 83 | \( 1 + 9.11T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 6.91T + 97T^{2} \) |
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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
−7.38994461031121956642871599622, −7.30299229980785341424858487116, −6.67892452174389750580776204959, −5.78040641329260006557218507305, −5.02832947363877595040718129775, −4.24326848942294284168840130482, −2.83919895121863082737995581876, −1.83118850833484500051265882160, −0.72761826427934126468404841149, 0,
0.72761826427934126468404841149, 1.83118850833484500051265882160, 2.83919895121863082737995581876, 4.24326848942294284168840130482, 5.02832947363877595040718129775, 5.78040641329260006557218507305, 6.67892452174389750580776204959, 7.30299229980785341424858487116, 7.38994461031121956642871599622