| L(s) = 1 | − 2-s + 3-s + 4-s + 4·5-s − 6-s − 2·7-s − 8-s + 9-s − 4·10-s + 11-s + 12-s − 4·13-s + 2·14-s + 4·15-s + 16-s + 2·17-s − 18-s + 4·20-s − 2·21-s − 22-s + 6·23-s − 24-s + 11·25-s + 4·26-s + 27-s − 2·28-s − 10·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.288·12-s − 1.10·13-s + 0.534·14-s + 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.894·20-s − 0.436·21-s − 0.213·22-s + 1.25·23-s − 0.204·24-s + 11/5·25-s + 0.784·26-s + 0.192·27-s − 0.377·28-s − 1.85·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.26451014349826, −13.45095733798823, −13.17561849606620, −12.85227816054947, −12.18439302156834, −11.64448801892237, −10.93821176206157, −10.34320734036756, −9.934399211401998, −9.664667944578190, −9.191486811175975, −8.895267960014875, −8.172362687140013, −7.503285866113204, −6.971306391488711, −6.608926693856364, −6.012072867380586, −5.432058134825786, −4.996019814642209, −4.136541195213913, −3.253342384644691, −2.764346973638247, −2.364342707950513, −1.593099591466211, −1.119589558493878, 0,
1.119589558493878, 1.593099591466211, 2.364342707950513, 2.764346973638247, 3.253342384644691, 4.136541195213913, 4.996019814642209, 5.432058134825786, 6.012072867380586, 6.608926693856364, 6.971306391488711, 7.503285866113204, 8.172362687140013, 8.895267960014875, 9.191486811175975, 9.664667944578190, 9.934399211401998, 10.34320734036756, 10.93821176206157, 11.64448801892237, 12.18439302156834, 12.85227816054947, 13.17561849606620, 13.45095733798823, 14.26451014349826
