| L(s) = 1 | − 19.1·2-s − 81·3-s − 146.·4-s + 625·5-s + 1.54e3·6-s − 2.40e3·7-s + 1.25e4·8-s + 6.56e3·9-s − 1.19e4·10-s + 6.69e4·11-s + 1.18e4·12-s − 1.13e5·13-s + 4.58e4·14-s − 5.06e4·15-s − 1.65e5·16-s + 5.03e5·17-s − 1.25e5·18-s − 6.66e5·19-s − 9.17e4·20-s + 1.94e5·21-s − 1.27e6·22-s − 2.33e5·23-s − 1.01e6·24-s + 3.90e5·25-s + 2.16e6·26-s − 5.31e5·27-s + 3.52e5·28-s + ⋯ |
| L(s) = 1 | − 0.844·2-s − 0.577·3-s − 0.286·4-s + 0.447·5-s + 0.487·6-s − 0.377·7-s + 1.08·8-s + 0.333·9-s − 0.377·10-s + 1.37·11-s + 0.165·12-s − 1.09·13-s + 0.319·14-s − 0.258·15-s − 0.631·16-s + 1.46·17-s − 0.281·18-s − 1.17·19-s − 0.128·20-s + 0.218·21-s − 1.16·22-s − 0.173·23-s − 0.627·24-s + 0.200·25-s + 0.928·26-s − 0.192·27-s + 0.108·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8350399990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8350399990\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 2 | \( 1 + 19.1T + 512T^{2} \) |
| 11 | \( 1 - 6.69e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.13e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.03e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 6.66e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.33e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.70e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.60e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.33e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.57e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.97e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 6.65e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.63e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.18e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 3.11e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.76e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.87e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.47e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.80e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.04e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.67e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.69e9T + 7.60e17T^{2} \) |
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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
−11.94221105956604710662524817668, −10.57211575687839843380207332284, −9.759184615089516068602960660019, −9.016011265072199760048413971389, −7.62376916492646723854130347302, −6.49706071587561980565655244627, −5.18955598356575558644080282844, −3.86455223318867957534343988710, −1.81034735931760690852168849796, −0.60729270156525285857178521744,
0.60729270156525285857178521744, 1.81034735931760690852168849796, 3.86455223318867957534343988710, 5.18955598356575558644080282844, 6.49706071587561980565655244627, 7.62376916492646723854130347302, 9.016011265072199760048413971389, 9.759184615089516068602960660019, 10.57211575687839843380207332284, 11.94221105956604710662524817668
