| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 7-s + 8-s + 9-s − 10-s + 1.46·11-s − 12-s + 3.46·13-s − 14-s + 15-s + 16-s − 2·17-s + 18-s + 4·19-s − 20-s + 21-s + 1.46·22-s − 23-s − 24-s + 25-s + 3.46·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.441·11-s − 0.288·12-s + 0.960·13-s − 0.267·14-s + 0.258·15-s + 0.250·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.218·21-s + 0.312·22-s − 0.208·23-s − 0.204·24-s + 0.200·25-s + 0.679·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.334493800\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.334493800\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 0.535T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 8.92T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 1.46T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 1.46T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 4.53T + 89T^{2} \) |
| 97 | \( 1 - 7.46T + 97T^{2} \) |
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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
−8.214854429590917058928578070892, −7.28993252083067400675174726552, −6.69725067817358637713557710299, −6.09658008915516018365370287642, −5.33615983004862243610337478141, −4.58068535907491009304192510202, −3.75187741203651232208766355462, −3.21312982175441756516514056250, −1.92668470134742143102605378909, −0.792773608342001275479717830155,
0.792773608342001275479717830155, 1.92668470134742143102605378909, 3.21312982175441756516514056250, 3.75187741203651232208766355462, 4.58068535907491009304192510202, 5.33615983004862243610337478141, 6.09658008915516018365370287642, 6.69725067817358637713557710299, 7.28993252083067400675174726552, 8.214854429590917058928578070892
