| L(s) = 1 | − 0.673·2-s + 0.897·3-s − 1.54·4-s − 0.604·6-s − 7-s + 2.38·8-s − 2.19·9-s − 1.69·11-s − 1.38·12-s − 0.185·13-s + 0.673·14-s + 1.48·16-s − 6.58·17-s + 1.47·18-s − 0.983·19-s − 0.897·21-s + 1.13·22-s − 23-s + 2.14·24-s + 0.124·26-s − 4.66·27-s + 1.54·28-s + 7.11·29-s − 4.91·31-s − 5.77·32-s − 1.51·33-s + 4.43·34-s + ⋯ |
| L(s) = 1 | − 0.476·2-s + 0.518·3-s − 0.773·4-s − 0.246·6-s − 0.377·7-s + 0.844·8-s − 0.731·9-s − 0.510·11-s − 0.400·12-s − 0.0514·13-s + 0.179·14-s + 0.371·16-s − 1.59·17-s + 0.348·18-s − 0.225·19-s − 0.195·21-s + 0.242·22-s − 0.208·23-s + 0.437·24-s + 0.0245·26-s − 0.897·27-s + 0.292·28-s + 1.32·29-s − 0.883·31-s − 1.02·32-s − 0.264·33-s + 0.759·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7574438906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7574438906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.673T + 2T^{2} \) |
| 3 | \( 1 - 0.897T + 3T^{2} \) |
| 11 | \( 1 + 1.69T + 11T^{2} \) |
| 13 | \( 1 + 0.185T + 13T^{2} \) |
| 17 | \( 1 + 6.58T + 17T^{2} \) |
| 19 | \( 1 + 0.983T + 19T^{2} \) |
| 29 | \( 1 - 7.11T + 29T^{2} \) |
| 31 | \( 1 + 4.91T + 31T^{2} \) |
| 37 | \( 1 + 1.58T + 37T^{2} \) |
| 41 | \( 1 - 0.327T + 41T^{2} \) |
| 43 | \( 1 - 6.87T + 43T^{2} \) |
| 47 | \( 1 - 10.1T + 47T^{2} \) |
| 53 | \( 1 + 5.51T + 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 3.44T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 0.255T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 0.304T + 89T^{2} \) |
| 97 | \( 1 + 5.81T + 97T^{2} \) |
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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
−8.549903257545620715557279737942, −7.960441992165585444036189944577, −7.17803857279391230911296406342, −6.26991491473073237178340144633, −5.43684883271431779203288354235, −4.58368223036132383796563984966, −3.86741583873483016965756984233, −2.85940566067785012316757544723, −2.04326321502514838604050693671, −0.50251093135465238965746592049,
0.50251093135465238965746592049, 2.04326321502514838604050693671, 2.85940566067785012316757544723, 3.86741583873483016965756984233, 4.58368223036132383796563984966, 5.43684883271431779203288354235, 6.26991491473073237178340144633, 7.17803857279391230911296406342, 7.960441992165585444036189944577, 8.549903257545620715557279737942
