| L(s) = 1 | + 19.8·2-s − 12.8·3-s + 266.·4-s − 254.·6-s − 239.·7-s + 2.74e3·8-s − 2.02e3·9-s + 609.·11-s − 3.41e3·12-s − 2.38e3·13-s − 4.76e3·14-s + 2.04e4·16-s − 8.50e3·17-s − 4.01e4·18-s + 5.28e4·19-s + 3.07e3·21-s + 1.21e4·22-s + 3.68e3·23-s − 3.51e4·24-s − 4.73e4·26-s + 5.39e4·27-s − 6.38e4·28-s + 1.52e5·29-s − 1.71e5·31-s + 5.47e4·32-s − 7.80e3·33-s − 1.68e5·34-s + ⋯ |
| L(s) = 1 | + 1.75·2-s − 0.273·3-s + 2.08·4-s − 0.480·6-s − 0.264·7-s + 1.89·8-s − 0.924·9-s + 0.138·11-s − 0.569·12-s − 0.300·13-s − 0.463·14-s + 1.24·16-s − 0.420·17-s − 1.62·18-s + 1.76·19-s + 0.0723·21-s + 0.242·22-s + 0.0631·23-s − 0.519·24-s − 0.527·26-s + 0.527·27-s − 0.549·28-s + 1.15·29-s − 1.03·31-s + 0.295·32-s − 0.0378·33-s − 0.737·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| good | 2 | \( 1 - 19.8T + 128T^{2} \) |
| 3 | \( 1 + 12.8T + 2.18e3T^{2} \) |
| 7 | \( 1 + 239.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 609.T + 1.94e7T^{2} \) |
| 13 | \( 1 + 2.38e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 8.50e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.28e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.68e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.52e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.71e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.87e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.37e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 7.00e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.00e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.29e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.40e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.98e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.83e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.01e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.35e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 7.36e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.51e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.46e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 9.80e6T + 8.07e13T^{2} \) |
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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
−9.105072451961441306795876381161, −7.85827957871774508293993770242, −6.83597129212928743693034522471, −6.12849586764260535372090719068, −5.26376599053039584337474316663, −4.65864750449416522280409694013, −3.34160170472536782216701936211, −2.90139227145795766449581499361, −1.56884537779835517453579888552, 0,
1.56884537779835517453579888552, 2.90139227145795766449581499361, 3.34160170472536782216701936211, 4.65864750449416522280409694013, 5.26376599053039584337474316663, 6.12849586764260535372090719068, 6.83597129212928743693034522471, 7.85827957871774508293993770242, 9.105072451961441306795876381161
