| L(s) = 1 | + 2-s + 4-s − 5-s − 3·7-s + 8-s − 10-s + 5·11-s + 2·13-s − 3·14-s + 16-s − 6·17-s + 2·19-s − 20-s + 5·22-s − 4·23-s + 25-s + 2·26-s − 3·28-s + 4·29-s − 4·31-s + 32-s − 6·34-s + 3·35-s + 7·37-s + 2·38-s − 40-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s + 0.353·8-s − 0.316·10-s + 1.50·11-s + 0.554·13-s − 0.801·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s − 0.223·20-s + 1.06·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.566·28-s + 0.742·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.507·35-s + 1.15·37-s + 0.324·38-s − 0.158·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.630447995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.630447995\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.998644435417560119883977548596, −7.08865991340691419508554070607, −6.45196641696460145204263040679, −6.22958710103922363575492351194, −5.16590032382428402513781542548, −4.05286520046376729631367894099, −3.93612357195171696480454198471, −2.99422637877982941714572705306, −2.02427605437778737655266819311, −0.76173955376549307829451425842,
0.76173955376549307829451425842, 2.02427605437778737655266819311, 2.99422637877982941714572705306, 3.93612357195171696480454198471, 4.05286520046376729631367894099, 5.16590032382428402513781542548, 6.22958710103922363575492351194, 6.45196641696460145204263040679, 7.08865991340691419508554070607, 7.998644435417560119883977548596
