| L(s) = 1 | − 1.29·2-s − 1.51·3-s − 0.329·4-s + 1.96·6-s + 2.88·7-s + 3.01·8-s − 0.691·9-s + 5.91·11-s + 0.500·12-s + 2.82·13-s − 3.72·14-s − 3.23·16-s − 2.25·17-s + 0.893·18-s + 4.30·19-s − 4.38·21-s − 7.64·22-s − 3.09·23-s − 4.57·24-s − 3.65·26-s + 5.60·27-s − 0.950·28-s − 5.99·29-s − 1.34·31-s − 1.84·32-s − 8.98·33-s + 2.91·34-s + ⋯ |
| L(s) = 1 | − 0.913·2-s − 0.877·3-s − 0.164·4-s + 0.801·6-s + 1.09·7-s + 1.06·8-s − 0.230·9-s + 1.78·11-s + 0.144·12-s + 0.784·13-s − 0.996·14-s − 0.808·16-s − 0.546·17-s + 0.210·18-s + 0.987·19-s − 0.956·21-s − 1.62·22-s − 0.646·23-s − 0.933·24-s − 0.716·26-s + 1.07·27-s − 0.179·28-s − 1.11·29-s − 0.240·31-s − 0.325·32-s − 1.56·33-s + 0.499·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 3 | \( 1 + 1.51T + 3T^{2} \) |
| 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 2.25T + 17T^{2} \) |
| 19 | \( 1 - 4.30T + 19T^{2} \) |
| 23 | \( 1 + 3.09T + 23T^{2} \) |
| 29 | \( 1 + 5.99T + 29T^{2} \) |
| 31 | \( 1 + 1.34T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 2.35T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 0.476T + 47T^{2} \) |
| 53 | \( 1 + 6.34T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 8.01T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 9.71T + 71T^{2} \) |
| 73 | \( 1 + 7.56T + 73T^{2} \) |
| 79 | \( 1 + 3.49T + 79T^{2} \) |
| 83 | \( 1 - 7.43T + 83T^{2} \) |
| 89 | \( 1 - 3.98T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.164021932385721722944278609348, −7.14947698082662138730179085043, −6.62663954554026893042129079018, −5.65822330675074277324793652412, −5.07183212008672946225819272406, −4.22713332662194653402422710695, −3.51088377459680543414182417246, −1.71472949812181859656132800615, −1.29931826457089701616564039439, 0,
1.29931826457089701616564039439, 1.71472949812181859656132800615, 3.51088377459680543414182417246, 4.22713332662194653402422710695, 5.07183212008672946225819272406, 5.65822330675074277324793652412, 6.62663954554026893042129079018, 7.14947698082662138730179085043, 8.164021932385721722944278609348
