L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 5·7-s + 8-s + 9-s + 10-s − 11-s − 12-s + 13-s + 5·14-s − 15-s + 16-s − 2·17-s + 18-s − 4·19-s + 20-s − 5·21-s − 22-s − 9·23-s − 24-s + 25-s + 26-s − 27-s + 5·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.88·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s − 1.09·21-s − 0.213·22-s − 1.87·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−12.73424322649077, −12.27126681534909, −11.91118373582323, −11.59710667328697, −11.07494710537239, −10.55902260709437, −10.38473293499595, −9.933681397666289, −9.023266491274610, −8.613404002746677, −8.148116545856873, −7.819365541570082, −7.191949558855708, −6.606230591710034, −6.275109301027204, −5.579802546597625, −5.376056277628222, −4.764880162500319, −4.405971337342258, −3.996812654468363, −3.304694297779064, −2.428150087412158, −1.894628413748443, −1.760952855758371, −0.9155121465608693, 0,
0.9155121465608693, 1.760952855758371, 1.894628413748443, 2.428150087412158, 3.304694297779064, 3.996812654468363, 4.405971337342258, 4.764880162500319, 5.376056277628222, 5.579802546597625, 6.275109301027204, 6.606230591710034, 7.191949558855708, 7.819365541570082, 8.148116545856873, 8.613404002746677, 9.023266491274610, 9.933681397666289, 10.38473293499595, 10.55902260709437, 11.07494710537239, 11.59710667328697, 11.91118373582323, 12.27126681534909, 12.73424322649077