L(s) = 1 | + 1.50·2-s − 3-s + 0.271·4-s − 5-s − 1.50·6-s − 7-s − 2.60·8-s + 9-s − 1.50·10-s + 2.60·11-s − 0.271·12-s − 2.83·13-s − 1.50·14-s + 15-s − 4.46·16-s + 4.57·17-s + 1.50·18-s − 0.659·19-s − 0.271·20-s + 21-s + 3.92·22-s + 23-s + 2.60·24-s + 25-s − 4.26·26-s − 27-s − 0.271·28-s + ⋯ |
L(s) = 1 | + 1.06·2-s − 0.577·3-s + 0.135·4-s − 0.447·5-s − 0.615·6-s − 0.377·7-s − 0.920·8-s + 0.333·9-s − 0.476·10-s + 0.786·11-s − 0.0785·12-s − 0.785·13-s − 0.402·14-s + 0.258·15-s − 1.11·16-s + 1.11·17-s + 0.355·18-s − 0.151·19-s − 0.0608·20-s + 0.218·21-s + 0.837·22-s + 0.208·23-s + 0.531·24-s + 0.200·25-s − 0.837·26-s − 0.192·27-s − 0.0513·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.783878882\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.783878882\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 1.50T + 2T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 17 | \( 1 - 4.57T + 17T^{2} \) |
| 19 | \( 1 + 0.659T + 19T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 - 8.42T + 31T^{2} \) |
| 37 | \( 1 + 3.37T + 37T^{2} \) |
| 41 | \( 1 - 7.83T + 41T^{2} \) |
| 43 | \( 1 - 12.0T + 43T^{2} \) |
| 47 | \( 1 - 9.01T + 47T^{2} \) |
| 53 | \( 1 + 5.60T + 53T^{2} \) |
| 59 | \( 1 + 8.77T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 - 7.98T + 67T^{2} \) |
| 71 | \( 1 - 6.63T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 + 4.25T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 8.61T + 89T^{2} \) |
| 97 | \( 1 - 8.49T + 97T^{2} \) |
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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
−9.261526052368250607273924792677, −7.988620683778637857115793294075, −7.23810972903887522767024790962, −6.38175817475512509346457257906, −5.71204201583698888701973657383, −4.99619304612251397410161853764, −4.11825698821944310413665899212, −3.56739021502311175519561526306, −2.46356460733964575322142628518, −0.74383651182353871892617623803,
0.74383651182353871892617623803, 2.46356460733964575322142628518, 3.56739021502311175519561526306, 4.11825698821944310413665899212, 4.99619304612251397410161853764, 5.71204201583698888701973657383, 6.38175817475512509346457257906, 7.23810972903887522767024790962, 7.988620683778637857115793294075, 9.261526052368250607273924792677