| L(s) = 1 | + 2-s + 3.10·3-s + 4-s + 5-s + 3.10·6-s − 7-s + 8-s + 6.62·9-s + 10-s + 11-s + 3.10·12-s − 3.62·13-s − 14-s + 3.10·15-s + 16-s − 4.20·17-s + 6.62·18-s − 8.15·19-s + 20-s − 3.10·21-s + 22-s + 0.897·23-s + 3.10·24-s + 25-s − 3.62·26-s + 11.2·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.79·3-s + 0.5·4-s + 0.447·5-s + 1.26·6-s − 0.377·7-s + 0.353·8-s + 2.20·9-s + 0.316·10-s + 0.301·11-s + 0.895·12-s − 1.00·13-s − 0.267·14-s + 0.801·15-s + 0.250·16-s − 1.01·17-s + 1.56·18-s − 1.87·19-s + 0.223·20-s − 0.677·21-s + 0.213·22-s + 0.187·23-s + 0.633·24-s + 0.200·25-s − 0.711·26-s + 2.16·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.093797967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.093797967\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 3.10T + 3T^{2} \) |
| 13 | \( 1 + 3.62T + 13T^{2} \) |
| 17 | \( 1 + 4.20T + 17T^{2} \) |
| 19 | \( 1 + 8.15T + 19T^{2} \) |
| 23 | \( 1 - 0.897T + 23T^{2} \) |
| 29 | \( 1 + 7.30T + 29T^{2} \) |
| 31 | \( 1 + 3.42T + 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 - 12.3T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 1.15T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 7.25T + 59T^{2} \) |
| 61 | \( 1 + 3.15T + 61T^{2} \) |
| 67 | \( 1 + 8.41T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 0.205T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 2.20T + 83T^{2} \) |
| 89 | \( 1 + 7.45T + 89T^{2} \) |
| 97 | \( 1 - 2.25T + 97T^{2} \) |
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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
−10.19804626987765361386041370629, −9.194817351196831512690578516945, −8.858451861415374889270035817489, −7.62554813657593880164831646119, −7.00127292544751390939920300063, −5.95234927606692060360042708197, −4.47881869210866016638093045778, −3.85259624764558711104415686920, −2.56862858462667886089264267108, −2.07917122607290327788046218850,
2.07917122607290327788046218850, 2.56862858462667886089264267108, 3.85259624764558711104415686920, 4.47881869210866016638093045778, 5.95234927606692060360042708197, 7.00127292544751390939920300063, 7.62554813657593880164831646119, 8.858451861415374889270035817489, 9.194817351196831512690578516945, 10.19804626987765361386041370629
