| L(s) = 1 | − 8·2-s + 64·4-s − 366.·5-s − 561.·7-s − 512·8-s + 2.93e3·10-s + 1.09e3·11-s + 6.92e3·13-s + 4.49e3·14-s + 4.09e3·16-s − 1.36e4·17-s + 2.95e4·19-s − 2.34e4·20-s − 8.76e3·22-s + 1.21e4·23-s + 5.64e4·25-s − 5.54e4·26-s − 3.59e4·28-s − 1.50e5·29-s − 2.04e5·31-s − 3.27e4·32-s + 1.09e5·34-s + 2.06e5·35-s − 4.92e5·37-s − 2.36e5·38-s + 1.87e5·40-s − 4.16e5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.31·5-s − 0.618·7-s − 0.353·8-s + 0.928·10-s + 0.248·11-s + 0.874·13-s + 0.437·14-s + 0.250·16-s − 0.674·17-s + 0.988·19-s − 0.656·20-s − 0.175·22-s + 0.208·23-s + 0.722·25-s − 0.618·26-s − 0.309·28-s − 1.14·29-s − 1.23·31-s − 0.176·32-s + 0.476·34-s + 0.812·35-s − 1.59·37-s − 0.698·38-s + 0.464·40-s − 0.943·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5422613752\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5422613752\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 5 | \( 1 + 366.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 561.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.09e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 6.92e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.36e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.95e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 1.50e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.04e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.92e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.71e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.30e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.01e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.28e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.95e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.46e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.82e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.03e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.36e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.27e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.88e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.46e6T + 8.07e13T^{2} \) |
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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
−9.951707721827234990394872270089, −9.027340942956244248978989578251, −8.294516588813821609587958829716, −7.33203233512248541191966874794, −6.65746002714455543858814411554, −5.36505121962281179666775553495, −3.86571579890463152458009615811, −3.26920542150947370117831459112, −1.65881023831069946929524344700, −0.37560953930429207363510728184,
0.37560953930429207363510728184, 1.65881023831069946929524344700, 3.26920542150947370117831459112, 3.86571579890463152458009615811, 5.36505121962281179666775553495, 6.65746002714455543858814411554, 7.33203233512248541191966874794, 8.294516588813821609587958829716, 9.027340942956244248978989578251, 9.951707721827234990394872270089
