| L(s) = 1 | − 2·2-s + 4·4-s − 5·5-s − 34·7-s − 8·8-s + 10·10-s + 48·11-s − 70·13-s + 68·14-s + 16·16-s + 27·17-s + 119·19-s − 20·20-s − 96·22-s + 51·23-s + 25·25-s + 140·26-s − 136·28-s + 30·29-s − 133·31-s − 32·32-s − 54·34-s + 170·35-s + 218·37-s − 238·38-s + 40·40-s − 156·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.83·7-s − 0.353·8-s + 0.316·10-s + 1.31·11-s − 1.49·13-s + 1.29·14-s + 1/4·16-s + 0.385·17-s + 1.43·19-s − 0.223·20-s − 0.930·22-s + 0.462·23-s + 1/5·25-s + 1.05·26-s − 0.917·28-s + 0.192·29-s − 0.770·31-s − 0.176·32-s − 0.272·34-s + 0.821·35-s + 0.968·37-s − 1.01·38-s + 0.158·40-s − 0.594·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8443230326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8443230326\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 13 | \( 1 + 70 T + p^{3} T^{2} \) |
| 17 | \( 1 - 27 T + p^{3} T^{2} \) |
| 19 | \( 1 - 119 T + p^{3} T^{2} \) |
| 23 | \( 1 - 51 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 + 133 T + p^{3} T^{2} \) |
| 37 | \( 1 - 218 T + p^{3} T^{2} \) |
| 41 | \( 1 + 156 T + p^{3} T^{2} \) |
| 43 | \( 1 + 88 T + p^{3} T^{2} \) |
| 47 | \( 1 - 516 T + p^{3} T^{2} \) |
| 53 | \( 1 - 639 T + p^{3} T^{2} \) |
| 59 | \( 1 - 654 T + p^{3} T^{2} \) |
| 61 | \( 1 - 461 T + p^{3} T^{2} \) |
| 67 | \( 1 - 182 T + p^{3} T^{2} \) |
| 71 | \( 1 + 900 T + p^{3} T^{2} \) |
| 73 | \( 1 - 704 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1375 T + p^{3} T^{2} \) |
| 83 | \( 1 + 915 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1116 T + p^{3} T^{2} \) |
| 97 | \( 1 + 16 T + p^{3} T^{2} \) |
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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
−11.64691622075633832467072738022, −10.14525322372882782241003599098, −9.613527744774681501175883847679, −8.880337951085025746680152962606, −7.31907772037916529507224891912, −6.89632644891457752547887438924, −5.61985136337645781212437179376, −3.83572262105792911258194543428, −2.78049431331423938905077552445, −0.71191097708518329911719741346,
0.71191097708518329911719741346, 2.78049431331423938905077552445, 3.83572262105792911258194543428, 5.61985136337645781212437179376, 6.89632644891457752547887438924, 7.31907772037916529507224891912, 8.880337951085025746680152962606, 9.613527744774681501175883847679, 10.14525322372882782241003599098, 11.64691622075633832467072738022
