L(s) = 1 | − 2.28·2-s + 0.432·3-s + 3.21·4-s − 5-s − 0.988·6-s − 7-s − 2.78·8-s − 2.81·9-s + 2.28·10-s − 2.32·11-s + 1.39·12-s + 2.28·14-s − 0.432·15-s − 0.0721·16-s − 7.11·17-s + 6.42·18-s + 4.06·19-s − 3.21·20-s − 0.432·21-s + 5.30·22-s + 6.76·23-s − 1.20·24-s + 25-s − 2.51·27-s − 3.21·28-s + 2.87·29-s + 0.988·30-s + ⋯ |
L(s) = 1 | − 1.61·2-s + 0.249·3-s + 1.60·4-s − 0.447·5-s − 0.403·6-s − 0.377·7-s − 0.985·8-s − 0.937·9-s + 0.722·10-s − 0.699·11-s + 0.402·12-s + 0.610·14-s − 0.111·15-s − 0.0180·16-s − 1.72·17-s + 1.51·18-s + 0.931·19-s − 0.719·20-s − 0.0944·21-s + 1.13·22-s + 1.40·23-s − 0.246·24-s + 0.200·25-s − 0.484·27-s − 0.608·28-s + 0.533·29-s + 0.180·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 3 | \( 1 - 0.432T + 3T^{2} \) |
| 11 | \( 1 + 2.32T + 11T^{2} \) |
| 17 | \( 1 + 7.11T + 17T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 23 | \( 1 - 6.76T + 23T^{2} \) |
| 29 | \( 1 - 2.87T + 29T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 37 | \( 1 - 2.45T + 37T^{2} \) |
| 41 | \( 1 - 8.12T + 41T^{2} \) |
| 43 | \( 1 + 3.68T + 43T^{2} \) |
| 47 | \( 1 - 4.69T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 - 9.98T + 59T^{2} \) |
| 61 | \( 1 - 0.735T + 61T^{2} \) |
| 67 | \( 1 - 5.11T + 67T^{2} \) |
| 71 | \( 1 + 4.09T + 71T^{2} \) |
| 73 | \( 1 + 5.17T + 73T^{2} \) |
| 79 | \( 1 - 6.45T + 79T^{2} \) |
| 83 | \( 1 + 3.43T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.947074424463792753339469493236, −7.17795861989588456114040980544, −6.79454058212864647343380935304, −5.76186070002842548617479219841, −4.93807765403599305337571436477, −3.85803651592106014793656279859, −2.76442836968282512132638827371, −2.34967997045486182922594664200, −0.935755884921625875877445971096, 0,
0.935755884921625875877445971096, 2.34967997045486182922594664200, 2.76442836968282512132638827371, 3.85803651592106014793656279859, 4.93807765403599305337571436477, 5.76186070002842548617479219841, 6.79454058212864647343380935304, 7.17795861989588456114040980544, 7.947074424463792753339469493236