| L(s) = 1 | − 9.27·3-s + 5.95·5-s − 7·7-s + 59.0·9-s + 11·11-s + 15.8·13-s − 55.2·15-s − 44.4·17-s − 142.·19-s + 64.9·21-s − 8.28·23-s − 89.5·25-s − 296.·27-s + 188.·29-s + 220.·31-s − 102.·33-s − 41.6·35-s + 156.·37-s − 146.·39-s + 64.8·41-s + 151.·43-s + 351.·45-s − 189.·47-s + 49·49-s + 412.·51-s − 19.1·53-s + 65.4·55-s + ⋯ |
| L(s) = 1 | − 1.78·3-s + 0.532·5-s − 0.377·7-s + 2.18·9-s + 0.301·11-s + 0.338·13-s − 0.950·15-s − 0.634·17-s − 1.71·19-s + 0.674·21-s − 0.0750·23-s − 0.716·25-s − 2.11·27-s + 1.20·29-s + 1.27·31-s − 0.538·33-s − 0.201·35-s + 0.695·37-s − 0.603·39-s + 0.247·41-s + 0.538·43-s + 1.16·45-s − 0.588·47-s + 0.142·49-s + 1.13·51-s − 0.0495·53-s + 0.160·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 + 9.27T + 27T^{2} \) |
| 5 | \( 1 - 5.95T + 125T^{2} \) |
| 13 | \( 1 - 15.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 142.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 8.28T + 1.21e4T^{2} \) |
| 29 | \( 1 - 188.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 156.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 64.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 151.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 189.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 19.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 306.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 387.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 95.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 294.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 676.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 912.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 236.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 854.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.60e3T + 9.12e5T^{2} \) |
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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
−9.109718732958259347743759146070, −8.073925106722734312916661502519, −6.73006885733328812709838185143, −6.39991241052710603627509024908, −5.77198787882701721085607848956, −4.71145446404979194842971755679, −4.06521302041563926912556724862, −2.34064403223413215077461014651, −1.08742501638657976948156490682, 0,
1.08742501638657976948156490682, 2.34064403223413215077461014651, 4.06521302041563926912556724862, 4.71145446404979194842971755679, 5.77198787882701721085607848956, 6.39991241052710603627509024908, 6.73006885733328812709838185143, 8.073925106722734312916661502519, 9.109718732958259347743759146070
