| L(s) = 1 | − 3·3-s + 5·7-s + 6·9-s + 4·11-s − 13-s + 3·17-s − 19-s − 15·21-s − 7·23-s − 9·27-s + 3·29-s − 2·31-s − 12·33-s − 2·37-s + 3·39-s − 6·41-s + 6·43-s + 18·49-s − 9·51-s − 13·53-s + 3·57-s + 9·59-s + 12·61-s + 30·63-s − 3·67-s + 21·69-s − 11·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.88·7-s + 2·9-s + 1.20·11-s − 0.277·13-s + 0.727·17-s − 0.229·19-s − 3.27·21-s − 1.45·23-s − 1.73·27-s + 0.557·29-s − 0.359·31-s − 2.08·33-s − 0.328·37-s + 0.480·39-s − 0.937·41-s + 0.914·43-s + 18/7·49-s − 1.26·51-s − 1.78·53-s + 0.397·57-s + 1.17·59-s + 1.53·61-s + 3.77·63-s − 0.366·67-s + 2.52·69-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−15.54486210175006, −14.63433132644096, −14.46287170268406, −13.98906924966434, −13.13938945732937, −12.36072976946466, −12.06185023330305, −11.57087786718316, −11.37346206805668, −10.65664321085503, −10.23265706072356, −9.665935712942339, −8.823966462442802, −8.198615706912924, −7.664948904933150, −7.014106398908248, −6.463794509759201, −5.776475804314183, −5.378821841827661, −4.756034452659343, −4.262794302377942, −3.724375056564692, −2.313225169004679, −1.470113140731210, −1.165308869985493, 0,
1.165308869985493, 1.470113140731210, 2.313225169004679, 3.724375056564692, 4.262794302377942, 4.756034452659343, 5.378821841827661, 5.776475804314183, 6.463794509759201, 7.014106398908248, 7.664948904933150, 8.198615706912924, 8.823966462442802, 9.665935712942339, 10.23265706072356, 10.65664321085503, 11.37346206805668, 11.57087786718316, 12.06185023330305, 12.36072976946466, 13.13938945732937, 13.98906924966434, 14.46287170268406, 14.63433132644096, 15.54486210175006
