| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s − 7.17·7-s + 8·8-s + 10·10-s − 40.4·11-s + 32.0·13-s − 14.3·14-s + 16·16-s − 122.·17-s + 100.·19-s + 20·20-s − 80.8·22-s + 23·23-s + 25·25-s + 64.0·26-s − 28.6·28-s − 76.9·29-s + 113.·31-s + 32·32-s − 244.·34-s − 35.8·35-s + 249.·37-s + 201.·38-s + 40·40-s − 355.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.387·7-s + 0.353·8-s + 0.316·10-s − 1.10·11-s + 0.683·13-s − 0.273·14-s + 0.250·16-s − 1.74·17-s + 1.21·19-s + 0.223·20-s − 0.783·22-s + 0.208·23-s + 0.200·25-s + 0.483·26-s − 0.193·28-s − 0.492·29-s + 0.654·31-s + 0.176·32-s − 1.23·34-s − 0.173·35-s + 1.10·37-s + 0.860·38-s + 0.158·40-s − 1.35·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 + 7.17T + 343T^{2} \) |
| 11 | \( 1 + 40.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 32.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 100.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 76.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 249.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 355.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 222.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 257.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 401.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 544.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 866.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 833.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 418.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 390.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 624.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 180.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 475.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 532.T + 9.12e5T^{2} \) |
| 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
−8.333115305665905992140232741327, −7.45933416750167611909978773837, −6.62182583491804291938517429278, −5.97422761896357605241364518142, −5.13419507660280456263014646502, −4.41479077961552383230949971704, −3.26700207348359803123624052902, −2.58111459218519506228231318391, −1.48422377321163434523518982182, 0,
1.48422377321163434523518982182, 2.58111459218519506228231318391, 3.26700207348359803123624052902, 4.41479077961552383230949971704, 5.13419507660280456263014646502, 5.97422761896357605241364518142, 6.62182583491804291938517429278, 7.45933416750167611909978773837, 8.333115305665905992140232741327
