| L(s) = 1 | − 8·2-s + 64·4-s − 206.·5-s + 1.61e3·7-s − 512·8-s + 1.65e3·10-s + 4.67e3·11-s + 6.67e3·13-s − 1.29e4·14-s + 4.09e3·16-s − 2.57e4·17-s + 2.23e4·19-s − 1.32e4·20-s − 3.74e4·22-s + 2.34e4·23-s − 3.54e4·25-s − 5.33e4·26-s + 1.03e5·28-s + 1.62e5·29-s − 2.33e5·31-s − 3.27e4·32-s + 2.05e5·34-s − 3.33e5·35-s + 3.08e5·37-s − 1.79e5·38-s + 1.05e5·40-s − 3.15e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.738·5-s + 1.78·7-s − 0.353·8-s + 0.522·10-s + 1.05·11-s + 0.842·13-s − 1.26·14-s + 0.250·16-s − 1.27·17-s + 0.748·19-s − 0.369·20-s − 0.749·22-s + 0.401·23-s − 0.454·25-s − 0.595·26-s + 0.891·28-s + 1.24·29-s − 1.40·31-s − 0.176·32-s + 0.898·34-s − 1.31·35-s + 1.00·37-s − 0.529·38-s + 0.261·40-s − 0.714·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.774781133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.774781133\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 206.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.61e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.67e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.67e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.57e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.23e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.34e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.62e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.33e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.08e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.15e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.22e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.95e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.69e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.12e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 9.03e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.52e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 7.03e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.06e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.67e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.07e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.44e6T + 8.07e13T^{2} \) |
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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
−11.34644517119591652439680869147, −10.87385812959481115803557566461, −9.253610625503939599589841849155, −8.425397072156784973064910797559, −7.66321959713807052981375622499, −6.49749491836962237844046250964, −4.91359335468457718754695908498, −3.77482282943520464080142625587, −1.90869262921359989314504858860, −0.882709618872004372502140113736,
0.882709618872004372502140113736, 1.90869262921359989314504858860, 3.77482282943520464080142625587, 4.91359335468457718754695908498, 6.49749491836962237844046250964, 7.66321959713807052981375622499, 8.425397072156784973064910797559, 9.253610625503939599589841849155, 10.87385812959481115803557566461, 11.34644517119591652439680869147
