| L(s) = 1 | − 1.53·2-s − 3-s + 0.364·4-s + 1.53·6-s + 1.68·7-s + 2.51·8-s + 9-s − 2.97·11-s − 0.364·12-s + 0.232·13-s − 2.59·14-s − 4.59·16-s + 7.45·17-s − 1.53·18-s + 0.753·19-s − 1.68·21-s + 4.57·22-s + 0.872·23-s − 2.51·24-s − 0.358·26-s − 27-s + 0.614·28-s + 6.87·29-s − 9.81·31-s + 2.03·32-s + 2.97·33-s − 11.4·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 0.577·3-s + 0.182·4-s + 0.627·6-s + 0.637·7-s + 0.889·8-s + 0.333·9-s − 0.897·11-s − 0.105·12-s + 0.0645·13-s − 0.692·14-s − 1.14·16-s + 1.80·17-s − 0.362·18-s + 0.172·19-s − 0.367·21-s + 0.976·22-s + 0.181·23-s − 0.513·24-s − 0.0702·26-s − 0.192·27-s + 0.116·28-s + 1.27·29-s − 1.76·31-s + 0.360·32-s + 0.518·33-s − 1.96·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7477164038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7477164038\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 + 2.97T + 11T^{2} \) |
| 13 | \( 1 - 0.232T + 13T^{2} \) |
| 17 | \( 1 - 7.45T + 17T^{2} \) |
| 19 | \( 1 - 0.753T + 19T^{2} \) |
| 23 | \( 1 - 0.872T + 23T^{2} \) |
| 29 | \( 1 - 6.87T + 29T^{2} \) |
| 31 | \( 1 + 9.81T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 + 5.27T + 43T^{2} \) |
| 47 | \( 1 + 8.56T + 47T^{2} \) |
| 53 | \( 1 - 5.97T + 53T^{2} \) |
| 59 | \( 1 + 3.85T + 59T^{2} \) |
| 61 | \( 1 + 4.39T + 61T^{2} \) |
| 67 | \( 1 - 1.79T + 67T^{2} \) |
| 71 | \( 1 + 4.37T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 7.37T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 9.47T + 97T^{2} \) |
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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
−9.327101396145697290108717413263, −8.332035494976553784805958492167, −7.78848927273147535561419028846, −7.25272016649434100813756474191, −6.01943987463469837082567904120, −5.18841129001209414658234994953, −4.52900891224976686769356985828, −3.20251081279167544972989090713, −1.76264489528467325352558284408, −0.74361059488048017420941187906,
0.74361059488048017420941187906, 1.76264489528467325352558284408, 3.20251081279167544972989090713, 4.52900891224976686769356985828, 5.18841129001209414658234994953, 6.01943987463469837082567904120, 7.25272016649434100813756474191, 7.78848927273147535561419028846, 8.332035494976553784805958492167, 9.327101396145697290108717413263
