| L(s) = 1 | − 0.793·2-s − 2.25·3-s − 1.37·4-s + 3.65·5-s + 1.78·6-s − 0.286·7-s + 2.67·8-s + 2.06·9-s − 2.90·10-s + 11-s + 3.08·12-s − 4.96·13-s + 0.227·14-s − 8.23·15-s + 0.618·16-s + 17-s − 1.64·18-s − 3.37·19-s − 5.01·20-s + 0.645·21-s − 0.793·22-s + 8.02·23-s − 6.02·24-s + 8.39·25-s + 3.93·26-s + 2.09·27-s + 0.392·28-s + ⋯ |
| L(s) = 1 | − 0.561·2-s − 1.29·3-s − 0.685·4-s + 1.63·5-s + 0.729·6-s − 0.108·7-s + 0.945·8-s + 0.689·9-s − 0.918·10-s + 0.301·11-s + 0.890·12-s − 1.37·13-s + 0.0608·14-s − 2.12·15-s + 0.154·16-s + 0.242·17-s − 0.386·18-s − 0.773·19-s − 1.12·20-s + 0.140·21-s − 0.169·22-s + 1.67·23-s − 1.22·24-s + 1.67·25-s + 0.772·26-s + 0.403·27-s + 0.0742·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7301697863\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7301697863\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 0.793T + 2T^{2} \) |
| 3 | \( 1 + 2.25T + 3T^{2} \) |
| 5 | \( 1 - 3.65T + 5T^{2} \) |
| 7 | \( 1 + 0.286T + 7T^{2} \) |
| 13 | \( 1 + 4.96T + 13T^{2} \) |
| 19 | \( 1 + 3.37T + 19T^{2} \) |
| 23 | \( 1 - 8.02T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 + 9.15T + 31T^{2} \) |
| 37 | \( 1 - 1.92T + 37T^{2} \) |
| 41 | \( 1 + 1.29T + 41T^{2} \) |
| 47 | \( 1 + 6.41T + 47T^{2} \) |
| 53 | \( 1 - 8.87T + 53T^{2} \) |
| 59 | \( 1 - 0.744T + 59T^{2} \) |
| 61 | \( 1 + 3.95T + 61T^{2} \) |
| 67 | \( 1 + 2.29T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 - 8.89T + 73T^{2} \) |
| 79 | \( 1 + 6.58T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
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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
−7.71131378684132759066849113698, −7.01599529047727906605467299977, −6.40932625091120475915271274665, −5.63060847708683575250384373133, −5.11925232670320225126820029356, −4.78507052544141974114103687629, −3.56772230478822332904864542361, −2.31729453138115903985051842357, −1.54054210096713524521605724008, −0.51393719727767176755125658661,
0.51393719727767176755125658661, 1.54054210096713524521605724008, 2.31729453138115903985051842357, 3.56772230478822332904864542361, 4.78507052544141974114103687629, 5.11925232670320225126820029356, 5.63060847708683575250384373133, 6.40932625091120475915271274665, 7.01599529047727906605467299977, 7.71131378684132759066849113698
