L(s) = 1 | − 2.41·2-s + 3.20·3-s + 3.85·4-s + 5-s − 7.76·6-s − 2.07·7-s − 4.47·8-s + 7.29·9-s − 2.41·10-s − 4.56·11-s + 12.3·12-s + 3.08·13-s + 5.02·14-s + 3.20·15-s + 3.12·16-s + 3.05·17-s − 17.6·18-s − 0.524·19-s + 3.85·20-s − 6.67·21-s + 11.0·22-s + 7.45·23-s − 14.3·24-s + 25-s − 7.45·26-s + 13.7·27-s − 8.00·28-s + ⋯ |
L(s) = 1 | − 1.71·2-s + 1.85·3-s + 1.92·4-s + 0.447·5-s − 3.16·6-s − 0.785·7-s − 1.58·8-s + 2.43·9-s − 0.764·10-s − 1.37·11-s + 3.56·12-s + 0.854·13-s + 1.34·14-s + 0.828·15-s + 0.780·16-s + 0.740·17-s − 4.15·18-s − 0.120·19-s + 0.860·20-s − 1.45·21-s + 2.35·22-s + 1.55·23-s − 2.93·24-s + 0.200·25-s − 1.46·26-s + 2.65·27-s − 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.980565428\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.980565428\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 1607 | \( 1 + T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 - 3.20T + 3T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 + 4.56T + 11T^{2} \) |
| 13 | \( 1 - 3.08T + 13T^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 19 | \( 1 + 0.524T + 19T^{2} \) |
| 23 | \( 1 - 7.45T + 23T^{2} \) |
| 29 | \( 1 - 2.84T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 - 5.05T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 - 7.79T + 43T^{2} \) |
| 47 | \( 1 - 2.35T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 + 1.36T + 59T^{2} \) |
| 61 | \( 1 - 4.00T + 61T^{2} \) |
| 67 | \( 1 - 3.62T + 67T^{2} \) |
| 71 | \( 1 + 9.13T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 - 2.32T + 79T^{2} \) |
| 83 | \( 1 + 6.26T + 83T^{2} \) |
| 89 | \( 1 - 6.63T + 89T^{2} \) |
| 97 | \( 1 + 6.71T + 97T^{2} \) |
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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
−7.970429415048605187369178049390, −7.55209163652292691448603767462, −6.90973333883620269486291008787, −6.15391289957969471026508624698, −5.07101507635262849761163175231, −3.84782933226584520208236313979, −2.92976929004825729988890971485, −2.67508040948752157209595483753, −1.72308374017096966095526991307, −0.842938425027911934586699327242,
0.842938425027911934586699327242, 1.72308374017096966095526991307, 2.67508040948752157209595483753, 2.92976929004825729988890971485, 3.84782933226584520208236313979, 5.07101507635262849761163175231, 6.15391289957969471026508624698, 6.90973333883620269486291008787, 7.55209163652292691448603767462, 7.970429415048605187369178049390