| L(s) = 1 | − 7.57·2-s − 78.3·3-s − 70.6·4-s + 593.·6-s − 343·7-s + 1.50e3·8-s + 3.95e3·9-s + 3.03e3·11-s + 5.53e3·12-s − 7.16e3·13-s + 2.59e3·14-s − 2.36e3·16-s − 1.16e4·17-s − 2.99e4·18-s − 3.24e4·19-s + 2.68e4·21-s − 2.30e4·22-s − 7.03e4·23-s − 1.17e5·24-s + 5.42e4·26-s − 1.38e5·27-s + 2.42e4·28-s − 9.74e4·29-s + 1.54e5·31-s − 1.74e5·32-s − 2.38e5·33-s + 8.80e4·34-s + ⋯ |
| L(s) = 1 | − 0.669·2-s − 1.67·3-s − 0.551·4-s + 1.12·6-s − 0.377·7-s + 1.03·8-s + 1.80·9-s + 0.688·11-s + 0.924·12-s − 0.903·13-s + 0.253·14-s − 0.144·16-s − 0.573·17-s − 1.21·18-s − 1.08·19-s + 0.633·21-s − 0.461·22-s − 1.20·23-s − 1.74·24-s + 0.605·26-s − 1.35·27-s + 0.208·28-s − 0.741·29-s + 0.934·31-s − 0.942·32-s − 1.15·33-s + 0.384·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1176581305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1176581305\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + 343T \) |
| good | 2 | \( 1 + 7.57T + 128T^{2} \) |
| 3 | \( 1 + 78.3T + 2.18e3T^{2} \) |
| 11 | \( 1 - 3.03e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.16e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.16e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.24e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.03e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 9.74e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.54e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.13e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.20e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.80e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.40e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.09e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.09e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 7.22e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.07e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.04e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.99e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.15e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.44e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.78e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.08e6T + 8.07e13T^{2} \) |
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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
−11.32083926475759005899734479679, −10.30186780079532602963588189055, −9.709174837963458463010574598656, −8.459159944143943784791919540289, −7.08442915163547081782528553612, −6.20539002875079427790263734736, −4.99380816036870408003959372704, −4.11039931585381248278899644923, −1.66453275104354112579009091795, −0.23115399426977303858437546438,
0.23115399426977303858437546438, 1.66453275104354112579009091795, 4.11039931585381248278899644923, 4.99380816036870408003959372704, 6.20539002875079427790263734736, 7.08442915163547081782528553612, 8.459159944143943784791919540289, 9.709174837963458463010574598656, 10.30186780079532602963588189055, 11.32083926475759005899734479679
