| L(s) = 1 | − 2-s − 0.0975·3-s + 4-s + 1.88·5-s + 0.0975·6-s − 3.81·7-s − 8-s − 2.99·9-s − 1.88·10-s + 11-s − 0.0975·12-s − 6.37·13-s + 3.81·14-s − 0.184·15-s + 16-s + 2.53·17-s + 2.99·18-s + 1.88·20-s + 0.371·21-s − 22-s − 4.56·23-s + 0.0975·24-s − 1.43·25-s + 6.37·26-s + 0.584·27-s − 3.81·28-s + 1.96·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0563·3-s + 0.5·4-s + 0.844·5-s + 0.0398·6-s − 1.44·7-s − 0.353·8-s − 0.996·9-s − 0.597·10-s + 0.301·11-s − 0.0281·12-s − 1.76·13-s + 1.01·14-s − 0.0475·15-s + 0.250·16-s + 0.615·17-s + 0.704·18-s + 0.422·20-s + 0.0811·21-s − 0.213·22-s − 0.952·23-s + 0.0199·24-s − 0.286·25-s + 1.25·26-s + 0.112·27-s − 0.720·28-s + 0.365·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5367967300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5367967300\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.0975T + 3T^{2} \) |
| 5 | \( 1 - 1.88T + 5T^{2} \) |
| 7 | \( 1 + 3.81T + 7T^{2} \) |
| 13 | \( 1 + 6.37T + 13T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 23 | \( 1 + 4.56T + 23T^{2} \) |
| 29 | \( 1 - 1.96T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 + 1.18T + 41T^{2} \) |
| 43 | \( 1 - 9.70T + 43T^{2} \) |
| 47 | \( 1 - 0.143T + 47T^{2} \) |
| 53 | \( 1 - 3.50T + 53T^{2} \) |
| 59 | \( 1 + 9.30T + 59T^{2} \) |
| 61 | \( 1 + 6.68T + 61T^{2} \) |
| 67 | \( 1 + 9.40T + 67T^{2} \) |
| 71 | \( 1 + 9.48T + 71T^{2} \) |
| 73 | \( 1 - 3.62T + 73T^{2} \) |
| 79 | \( 1 + 0.848T + 79T^{2} \) |
| 83 | \( 1 + 2.01T + 83T^{2} \) |
| 89 | \( 1 + 2.48T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−7.66201053698748361543647693530, −7.32503584240398960568460106607, −6.34896353595094079958157856931, −5.92329018043021983096818799544, −5.39570439591386943077802810815, −4.22423875884215896288265915827, −3.16099048701039396510788999764, −2.63468744016446713355793153454, −1.83872562408121747729396225694, −0.37855076940530490761980925689,
0.37855076940530490761980925689, 1.83872562408121747729396225694, 2.63468744016446713355793153454, 3.16099048701039396510788999764, 4.22423875884215896288265915827, 5.39570439591386943077802810815, 5.92329018043021983096818799544, 6.34896353595094079958157856931, 7.32503584240398960568460106607, 7.66201053698748361543647693530
