L(s) = 1 | + 1.33·2-s + 6.37·3-s − 6.21·4-s − 5·5-s + 8.51·6-s + 16.1·7-s − 18.9·8-s + 13.6·9-s − 6.68·10-s + 11·11-s − 39.6·12-s − 55.6·13-s + 21.6·14-s − 31.8·15-s + 24.3·16-s − 8.16·17-s + 18.1·18-s − 19·19-s + 31.0·20-s + 103.·21-s + 14.6·22-s + 158.·23-s − 121.·24-s + 25·25-s − 74.3·26-s − 85.3·27-s − 100.·28-s + ⋯ |
L(s) = 1 | + 0.472·2-s + 1.22·3-s − 0.776·4-s − 0.447·5-s + 0.579·6-s + 0.873·7-s − 0.839·8-s + 0.504·9-s − 0.211·10-s + 0.301·11-s − 0.952·12-s − 1.18·13-s + 0.412·14-s − 0.548·15-s + 0.380·16-s − 0.116·17-s + 0.238·18-s − 0.229·19-s + 0.347·20-s + 1.07·21-s + 0.142·22-s + 1.43·23-s − 1.02·24-s + 0.200·25-s − 0.560·26-s − 0.608·27-s − 0.678·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
good | 2 | \( 1 - 1.33T + 8T^{2} \) |
| 3 | \( 1 - 6.37T + 27T^{2} \) |
| 7 | \( 1 - 16.1T + 343T^{2} \) |
| 13 | \( 1 + 55.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 8.16T + 4.91e3T^{2} \) |
| 23 | \( 1 - 158.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 190.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 375.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 514.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 25.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 236.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 183.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 772.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 390.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 525.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 693.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.29e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 660.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 35.3T + 9.12e5T^{2} \) |
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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
−8.874272328956024720503058826803, −8.491011733194628755250323628299, −7.64702966939760911436017683215, −6.77818764350612703563200675492, −5.18790823067396372346521133678, −4.76113275005401200930766950096, −3.65486167338021021833923567125, −2.95626567585277648894399762743, −1.68666908158714440736661985595, 0,
1.68666908158714440736661985595, 2.95626567585277648894399762743, 3.65486167338021021833923567125, 4.76113275005401200930766950096, 5.18790823067396372346521133678, 6.77818764350612703563200675492, 7.64702966939760911436017683215, 8.491011733194628755250323628299, 8.874272328956024720503058826803