| L(s) = 1 | + 3.56·2-s + 4.68·4-s + 6.06·7-s − 11.8·8-s − 61.3·11-s + 70.8·13-s + 21.5·14-s − 79.5·16-s − 51.5·17-s + 15.7·19-s − 218.·22-s − 114.·23-s + 252.·26-s + 28.4·28-s − 39.7·29-s − 240.·31-s − 188.·32-s − 183.·34-s + 129.·37-s + 56.2·38-s − 262.·41-s − 127.·43-s − 287.·44-s − 409.·46-s − 85.2·47-s − 306.·49-s + 331.·52-s + ⋯ |
| L(s) = 1 | + 1.25·2-s + 0.585·4-s + 0.327·7-s − 0.521·8-s − 1.68·11-s + 1.51·13-s + 0.412·14-s − 1.24·16-s − 0.735·17-s + 0.190·19-s − 2.11·22-s − 1.04·23-s + 1.90·26-s + 0.191·28-s − 0.254·29-s − 1.39·31-s − 1.04·32-s − 0.926·34-s + 0.575·37-s + 0.240·38-s − 0.999·41-s − 0.452·43-s − 0.985·44-s − 1.31·46-s − 0.264·47-s − 0.892·49-s + 0.885·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3.56T + 8T^{2} \) |
| 7 | \( 1 - 6.06T + 343T^{2} \) |
| 11 | \( 1 + 61.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 70.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 15.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 39.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 240.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 129.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 262.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 85.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 687.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 455.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 955.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 160.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 702.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 65.3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 996.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.33e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
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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
−9.772519987178082221682739036044, −8.627421272405659061861775279441, −7.944120685438689408996259576869, −6.69256049668903689888540579166, −5.75775042897308743253722896866, −5.11395637343069579937174399127, −4.08617185985768021584747718268, −3.18169044375320892871913009447, −1.98276231883436941238507234363, 0,
1.98276231883436941238507234363, 3.18169044375320892871913009447, 4.08617185985768021584747718268, 5.11395637343069579937174399127, 5.75775042897308743253722896866, 6.69256049668903689888540579166, 7.944120685438689408996259576869, 8.627421272405659061861775279441, 9.772519987178082221682739036044
