| L(s) = 1 | + 7.13·2-s + 27·3-s − 77.0·4-s + 192.·6-s + 112.·7-s − 1.46e3·8-s + 729·9-s + 656.·11-s − 2.08e3·12-s − 7.80e3·13-s + 805.·14-s − 575.·16-s − 1.46e4·17-s + 5.20e3·18-s − 4.25e4·19-s + 3.04e3·21-s + 4.68e3·22-s − 5.31e4·23-s − 3.95e4·24-s − 5.56e4·26-s + 1.96e4·27-s − 8.70e3·28-s − 1.67e5·29-s + 1.08e5·31-s + 1.83e5·32-s + 1.77e4·33-s − 1.04e5·34-s + ⋯ |
| L(s) = 1 | + 0.630·2-s + 0.577·3-s − 0.602·4-s + 0.364·6-s + 0.124·7-s − 1.01·8-s + 0.333·9-s + 0.148·11-s − 0.347·12-s − 0.985·13-s + 0.0784·14-s − 0.0351·16-s − 0.722·17-s + 0.210·18-s − 1.42·19-s + 0.0718·21-s + 0.0937·22-s − 0.910·23-s − 0.583·24-s − 0.621·26-s + 0.192·27-s − 0.0749·28-s − 1.27·29-s + 0.651·31-s + 0.988·32-s + 0.0858·33-s − 0.455·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 7.13T + 128T^{2} \) |
| 7 | \( 1 - 112.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 656.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.80e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.46e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.25e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.31e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.67e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.08e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.24e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 6.34e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.44e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.00e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 9.34e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 8.42e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.67e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.22e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.04e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.39e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.77e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.48e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.28e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.96e6T + 8.07e13T^{2} \) |
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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
−12.84941829857426625178435627891, −11.74476451547060353048354266782, −10.13674727801174657776992274864, −9.077702120540965566695246899454, −8.028500625939717141575204089163, −6.43225729300978571403629594816, −4.88479577188490182659343609434, −3.84313931145968140498812204120, −2.26290673483007228371179917995, 0,
2.26290673483007228371179917995, 3.84313931145968140498812204120, 4.88479577188490182659343609434, 6.43225729300978571403629594816, 8.028500625939717141575204089163, 9.077702120540965566695246899454, 10.13674727801174657776992274864, 11.74476451547060353048354266782, 12.84941829857426625178435627891
