| L(s) = 1 | + 0.618·2-s − 2.61·3-s − 1.61·4-s + 1.61·5-s − 1.61·6-s − 3·7-s − 2.23·8-s + 3.85·9-s + 1.00·10-s − 0.236·11-s + 4.23·12-s − 3·13-s − 1.85·14-s − 4.23·15-s + 1.85·16-s − 5.61·17-s + 2.38·18-s − 3·19-s − 2.61·20-s + 7.85·21-s − 0.145·22-s − 2.76·23-s + 5.85·24-s − 2.38·25-s − 1.85·26-s − 2.23·27-s + 4.85·28-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 1.51·3-s − 0.809·4-s + 0.723·5-s − 0.660·6-s − 1.13·7-s − 0.790·8-s + 1.28·9-s + 0.316·10-s − 0.0711·11-s + 1.22·12-s − 0.832·13-s − 0.495·14-s − 1.09·15-s + 0.463·16-s − 1.36·17-s + 0.561·18-s − 0.688·19-s − 0.585·20-s + 1.71·21-s − 0.0311·22-s − 0.576·23-s + 1.19·24-s − 0.476·25-s − 0.363·26-s − 0.430·27-s + 0.917·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 6007 | \( 1+O(T) \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 0.236T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 + 5.61T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 + 7.85T + 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 7.47T + 47T^{2} \) |
| 53 | \( 1 - 3.61T + 53T^{2} \) |
| 59 | \( 1 + 6.09T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 - 3T + 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 8.32T + 79T^{2} \) |
| 83 | \( 1 - 9.18T + 83T^{2} \) |
| 89 | \( 1 - 8.47T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−6.93152215157620671972099304000, −6.37590268278164801145706367589, −5.96645525718879330433819561574, −5.23537144989204326843621969774, −4.65365219003909139039858167905, −3.93205549651790109932233110751, −2.88539456395357843272259592611, −1.76219402088171842599433503801, 0, 0,
1.76219402088171842599433503801, 2.88539456395357843272259592611, 3.93205549651790109932233110751, 4.65365219003909139039858167905, 5.23537144989204326843621969774, 5.96645525718879330433819561574, 6.37590268278164801145706367589, 6.93152215157620671972099304000
