| L(s) = 1 | − 0.209·2-s + 1.33·3-s − 0.956·4-s + 0.618·5-s − 0.279·6-s − 7-s + 0.408·8-s + 0.790·9-s − 0.129·10-s + 11-s − 1.27·12-s + 0.209·14-s + 0.827·15-s + 0.870·16-s + 1.82·17-s − 0.165·18-s − 0.591·20-s − 1.33·21-s − 0.209·22-s + 0.547·24-s − 0.618·25-s − 0.279·27-s + 0.956·28-s − 1.61·29-s − 0.172·30-s − 0.591·32-s + 1.33·33-s + ⋯ |
| L(s) = 1 | − 0.209·2-s + 1.33·3-s − 0.956·4-s + 0.618·5-s − 0.279·6-s − 7-s + 0.408·8-s + 0.790·9-s − 0.129·10-s + 11-s − 1.27·12-s + 0.209·14-s + 0.827·15-s + 0.870·16-s + 1.82·17-s − 0.165·18-s − 0.591·20-s − 1.33·21-s − 0.209·22-s + 0.547·24-s − 0.618·25-s − 0.279·27-s + 0.956·28-s − 1.61·29-s − 0.172·30-s − 0.591·32-s + 1.33·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.108411890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108411890\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.209T + T^{2} \) |
| 3 | \( 1 - 1.33T + T^{2} \) |
| 5 | \( 1 - 0.618T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - 1.82T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.61T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + 1.95T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.33T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - 1.82T + T^{2} \) |
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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
−9.996248282830971219515488415667, −9.739008883657471176999576893218, −9.160500844371177573737514270941, −8.294055537533217198772719239564, −7.51849809618830232936191794820, −6.27968185269492516335198162885, −5.26946679224281481024992077457, −3.72847205514409784552957697124, −3.33115848203975470763211765789, −1.68895068151972648536867414180,
1.68895068151972648536867414180, 3.33115848203975470763211765789, 3.72847205514409784552957697124, 5.26946679224281481024992077457, 6.27968185269492516335198162885, 7.51849809618830232936191794820, 8.294055537533217198772719239564, 9.160500844371177573737514270941, 9.739008883657471176999576893218, 9.996248282830971219515488415667
