| L(s) = 1 | + 37.5·2-s + 901.·4-s + 205.·5-s + 7.43e3·7-s + 1.46e4·8-s + 7.71e3·10-s − 3.02e4·11-s − 1.47e5·13-s + 2.79e5·14-s + 8.88e4·16-s + 1.41e5·17-s − 2.97e5·19-s + 1.85e5·20-s − 1.13e6·22-s − 1.01e6·23-s − 1.91e6·25-s − 5.55e6·26-s + 6.70e6·28-s − 3.61e6·29-s + 2.87e5·31-s − 4.15e6·32-s + 5.31e6·34-s + 1.52e6·35-s − 7.45e6·37-s − 1.12e7·38-s + 3.00e6·40-s + 1.37e7·41-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.76·4-s + 0.146·5-s + 1.17·7-s + 1.26·8-s + 0.244·10-s − 0.622·11-s − 1.43·13-s + 1.94·14-s + 0.338·16-s + 0.410·17-s − 0.524·19-s + 0.258·20-s − 1.03·22-s − 0.757·23-s − 0.978·25-s − 2.38·26-s + 2.06·28-s − 0.950·29-s + 0.0559·31-s − 0.700·32-s + 0.682·34-s + 0.171·35-s − 0.653·37-s − 0.871·38-s + 0.185·40-s + 0.760·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 - 37.5T + 512T^{2} \) |
| 5 | \( 1 - 205.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.43e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.02e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.47e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.41e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.97e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.01e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.61e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.87e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 7.45e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.37e7T + 3.27e14T^{2} \) |
| 47 | \( 1 + 2.60e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.82e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.03e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.20e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.55e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.21e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.71e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.52e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.25e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.81e7T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.42e9T + 7.60e17T^{2} \) |
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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
−9.513610896334054381328343265127, −8.019924376389981869052231303314, −7.38290919471289332824759401729, −6.14248913840672686756386066883, −5.24296877759809770178751937058, −4.69967247168261489801371574889, −3.69679622428583599209902323215, −2.44415919924065089531098397056, −1.82079883852989318535015293966, 0,
1.82079883852989318535015293966, 2.44415919924065089531098397056, 3.69679622428583599209902323215, 4.69967247168261489801371574889, 5.24296877759809770178751937058, 6.14248913840672686756386066883, 7.38290919471289332824759401729, 8.019924376389981869052231303314, 9.513610896334054381328343265127
