| L(s) = 1 | − 2-s + 0.695·3-s + 4-s − 0.922·5-s − 0.695·6-s + 0.361·7-s − 8-s − 2.51·9-s + 0.922·10-s − 11-s + 0.695·12-s − 3.41·13-s − 0.361·14-s − 0.641·15-s + 16-s + 0.394·17-s + 2.51·18-s − 0.922·20-s + 0.251·21-s + 22-s − 8.03·23-s − 0.695·24-s − 4.14·25-s + 3.41·26-s − 3.83·27-s + 0.361·28-s − 9.79·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.401·3-s + 0.5·4-s − 0.412·5-s − 0.283·6-s + 0.136·7-s − 0.353·8-s − 0.838·9-s + 0.291·10-s − 0.301·11-s + 0.200·12-s − 0.946·13-s − 0.0966·14-s − 0.165·15-s + 0.250·16-s + 0.0956·17-s + 0.593·18-s − 0.206·20-s + 0.0548·21-s + 0.213·22-s − 1.67·23-s − 0.141·24-s − 0.829·25-s + 0.668·26-s − 0.738·27-s + 0.0683·28-s − 1.81·29-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
| \(L(1)\) |
\(\approx\) |
\(0.5988411893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5988411893\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.695T + 3T^{2} \) |
| 5 | \( 1 + 0.922T + 5T^{2} \) |
| 7 | \( 1 - 0.361T + 7T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 0.394T + 17T^{2} \) |
| 23 | \( 1 + 8.03T + 23T^{2} \) |
| 29 | \( 1 + 9.79T + 29T^{2} \) |
| 31 | \( 1 - 0.987T + 31T^{2} \) |
| 37 | \( 1 + 9.75T + 37T^{2} \) |
| 41 | \( 1 + 0.473T + 41T^{2} \) |
| 43 | \( 1 + 4.94T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 - 5.30T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 - 2.94T + 79T^{2} \) |
| 83 | \( 1 + 8.43T + 83T^{2} \) |
| 89 | \( 1 - 6.48T + 89T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
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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
−8.031109957065223803166389631816, −7.33501666682782477365857440910, −6.72511368021989168182912992831, −5.57611570659127455389005925445, −5.38094071272319718374822360796, −3.99144755150955465372296296781, −3.54243006707877659213815961996, −2.35483076326657674772814912955, −2.01117215929650084401577053621, −0.38874091002499552831149092781,
0.38874091002499552831149092781, 2.01117215929650084401577053621, 2.35483076326657674772814912955, 3.54243006707877659213815961996, 3.99144755150955465372296296781, 5.38094071272319718374822360796, 5.57611570659127455389005925445, 6.72511368021989168182912992831, 7.33501666682782477365857440910, 8.031109957065223803166389631816
