| L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 2·6-s + 3·7-s + 9-s − 5·11-s + 2·12-s + 13-s + 6·14-s − 4·16-s − 5·17-s + 2·18-s + 3·21-s − 10·22-s + 23-s + 2·26-s + 27-s + 6·28-s − 10·29-s + 2·31-s − 8·32-s − 5·33-s − 10·34-s + 2·36-s − 3·37-s + 39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1.13·7-s + 1/3·9-s − 1.50·11-s + 0.577·12-s + 0.277·13-s + 1.60·14-s − 16-s − 1.21·17-s + 0.471·18-s + 0.654·21-s − 2.13·22-s + 0.208·23-s + 0.392·26-s + 0.192·27-s + 1.13·28-s − 1.85·29-s + 0.359·31-s − 1.41·32-s − 0.870·33-s − 1.71·34-s + 1/3·36-s − 0.493·37-s + 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 351975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.852879807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.852879807\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−12.80112745052234, −12.22921897731782, −11.69674953181382, −11.20429238582209, −10.91689976552204, −10.55632447947197, −9.863790649545607, −9.187277946939322, −8.902596010412610, −8.388002576218213, −7.793937663111217, −7.516975409230467, −7.014258721085052, −6.339254881036309, −5.820668132570775, −5.391973860977773, −4.945076518905391, −4.472585906594823, −4.138487291594142, −3.540951563451646, −2.851795595267761, −2.567768803739728, −1.941105666721035, −1.504793387944356, −0.3401992161952798,
0.3401992161952798, 1.504793387944356, 1.941105666721035, 2.567768803739728, 2.851795595267761, 3.540951563451646, 4.138487291594142, 4.472585906594823, 4.945076518905391, 5.391973860977773, 5.820668132570775, 6.339254881036309, 7.014258721085052, 7.516975409230467, 7.793937663111217, 8.388002576218213, 8.902596010412610, 9.187277946939322, 9.863790649545607, 10.55632447947197, 10.91689976552204, 11.20429238582209, 11.69674953181382, 12.22921897731782, 12.80112745052234
