| L(s) = 1 | − 19.7·2-s − 123.·4-s − 1.55e3·5-s − 8.32e3·7-s + 1.25e4·8-s + 3.06e4·10-s − 3.07e4·11-s + 2.85e4·13-s + 1.64e5·14-s − 1.83e5·16-s + 6.37e5·17-s + 1.05e5·19-s + 1.91e5·20-s + 6.06e5·22-s + 5.11e5·23-s + 4.66e5·25-s − 5.63e5·26-s + 1.02e6·28-s − 7.81e5·29-s − 2.83e6·31-s − 2.78e6·32-s − 1.25e7·34-s + 1.29e7·35-s + 1.22e7·37-s − 2.08e6·38-s − 1.94e7·40-s + 6.83e6·41-s + ⋯ |
| L(s) = 1 | − 0.871·2-s − 0.240·4-s − 1.11·5-s − 1.31·7-s + 1.08·8-s + 0.969·10-s − 0.633·11-s + 0.277·13-s + 1.14·14-s − 0.701·16-s + 1.85·17-s + 0.186·19-s + 0.267·20-s + 0.551·22-s + 0.380·23-s + 0.238·25-s − 0.241·26-s + 0.315·28-s − 0.205·29-s − 0.550·31-s − 0.469·32-s − 1.61·34-s + 1.45·35-s + 1.07·37-s − 0.162·38-s − 1.20·40-s + 0.377·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - 2.85e4T \) |
| good | 2 | \( 1 + 19.7T + 512T^{2} \) |
| 5 | \( 1 + 1.55e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 8.32e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.07e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 6.37e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.05e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 5.11e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 7.81e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 2.83e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.22e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 6.83e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.84e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.30e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.42e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.63e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.90e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.22e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.65e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.42e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.64e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.46e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.65e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.98e7T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.98913497663874762533558695292, −9.995720884691401290877550072996, −9.183855776726886131378760257106, −7.942334096476252256670914178155, −7.34479932770188704101614407882, −5.69805649350041194026837434644, −4.12075583347254689604654219873, −3.07217613623678968420514448092, −0.946924238943557130341681988103, 0,
0.946924238943557130341681988103, 3.07217613623678968420514448092, 4.12075583347254689604654219873, 5.69805649350041194026837434644, 7.34479932770188704101614407882, 7.942334096476252256670914178155, 9.183855776726886131378760257106, 9.995720884691401290877550072996, 10.98913497663874762533558695292
