| L(s) = 1 | − 3.18·2-s + 2.13·4-s + 7·7-s + 18.6·8-s + 68.7·11-s − 56.2·13-s − 22.2·14-s − 76.5·16-s + 37.9·17-s − 26.0·19-s − 218.·22-s + 25.5·23-s + 178.·26-s + 14.9·28-s − 148.·29-s + 75.7·31-s + 94.0·32-s − 120.·34-s − 120.·37-s + 82.8·38-s − 345.·41-s − 287.·43-s + 146.·44-s − 81.2·46-s − 528.·47-s + 49·49-s − 119.·52-s + ⋯ |
| L(s) = 1 | − 1.12·2-s + 0.266·4-s + 0.377·7-s + 0.825·8-s + 1.88·11-s − 1.19·13-s − 0.425·14-s − 1.19·16-s + 0.541·17-s − 0.314·19-s − 2.11·22-s + 0.231·23-s + 1.34·26-s + 0.100·28-s − 0.949·29-s + 0.439·31-s + 0.519·32-s − 0.609·34-s − 0.534·37-s + 0.353·38-s − 1.31·41-s − 1.02·43-s + 0.501·44-s − 0.260·46-s − 1.64·47-s + 0.142·49-s − 0.319·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 3.18T + 8T^{2} \) |
| 11 | \( 1 - 68.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 56.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 37.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 25.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 75.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 120.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 345.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 287.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 528.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 361.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 705.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 393.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 591.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 668.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 251.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 295.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 916.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 736.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 142.T + 9.12e5T^{2} \) |
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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
−8.699055366280892698167572558277, −8.093750447463976233203695027528, −7.12847842894797679478698759031, −6.64570397061587375159867797273, −5.30231946914280291757442815637, −4.46899092041715832860235168364, −3.51964244420487138624493244320, −1.98627749954385164426234009923, −1.21709238854854974698756019739, 0,
1.21709238854854974698756019739, 1.98627749954385164426234009923, 3.51964244420487138624493244320, 4.46899092041715832860235168364, 5.30231946914280291757442815637, 6.64570397061587375159867797273, 7.12847842894797679478698759031, 8.093750447463976233203695027528, 8.699055366280892698167572558277
