| L(s) = 1 | − 2·2-s + 4·4-s − 10.8·5-s + 7·7-s − 8·8-s + 21.7·10-s − 21.7·11-s − 3.61·13-s − 14·14-s + 16·16-s − 11.2·17-s + 38·19-s − 43.4·20-s + 43.4·22-s − 37.8·23-s − 6.83·25-s + 7.22·26-s + 28·28-s + 80.5·29-s + 43.6·31-s − 32·32-s + 22.5·34-s − 76.0·35-s + 302.·37-s − 76·38-s + 86.9·40-s − 1.90·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.972·5-s + 0.377·7-s − 0.353·8-s + 0.687·10-s − 0.595·11-s − 0.0770·13-s − 0.267·14-s + 0.250·16-s − 0.160·17-s + 0.458·19-s − 0.486·20-s + 0.421·22-s − 0.343·23-s − 0.0546·25-s + 0.0544·26-s + 0.188·28-s + 0.515·29-s + 0.252·31-s − 0.176·32-s + 0.113·34-s − 0.367·35-s + 1.34·37-s − 0.324·38-s + 0.343·40-s − 0.00726·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9628412543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9628412543\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 10.8T + 125T^{2} \) |
| 11 | \( 1 + 21.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 3.61T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 38T + 6.85e3T^{2} \) |
| 23 | \( 1 + 37.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 80.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 43.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 302.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 1.90T + 6.89e4T^{2} \) |
| 43 | \( 1 - 78.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 258.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 106.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 383.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 672.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 89.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 707.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 492.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 675.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 996.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.42e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
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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
−10.94943475247315363414548408486, −10.04870325773691834800635207779, −9.016980444054066244244869796393, −7.976607828607942163687322129021, −7.59266165160191966481648284786, −6.34413923601211480663556764457, −5.02524174957327622557384038569, −3.79678895728298583283685696406, −2.41491107411319086662718772397, −0.71021349406902779112747883911,
0.71021349406902779112747883911, 2.41491107411319086662718772397, 3.79678895728298583283685696406, 5.02524174957327622557384038569, 6.34413923601211480663556764457, 7.59266165160191966481648284786, 7.976607828607942163687322129021, 9.016980444054066244244869796393, 10.04870325773691834800635207779, 10.94943475247315363414548408486
