| L(s) = 1 | + 8.23·3-s − 8.08·5-s − 7·7-s + 40.7·9-s + 14.7·11-s − 43.1·13-s − 66.5·15-s + 44.6·17-s + 2.31·19-s − 57.6·21-s − 169.·23-s − 59.7·25-s + 113.·27-s − 19.5·29-s − 173.·31-s + 121.·33-s + 56.5·35-s − 80.6·37-s − 355.·39-s − 41·41-s + 313.·43-s − 329.·45-s − 441.·47-s + 49·49-s + 367.·51-s + 294.·53-s − 119.·55-s + ⋯ |
| L(s) = 1 | + 1.58·3-s − 0.722·5-s − 0.377·7-s + 1.50·9-s + 0.405·11-s − 0.920·13-s − 1.14·15-s + 0.636·17-s + 0.0279·19-s − 0.598·21-s − 1.53·23-s − 0.477·25-s + 0.807·27-s − 0.125·29-s − 1.00·31-s + 0.641·33-s + 0.273·35-s − 0.358·37-s − 1.45·39-s − 0.156·41-s + 1.11·43-s − 1.09·45-s − 1.36·47-s + 0.142·49-s + 1.00·51-s + 0.764·53-s − 0.292·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 - 8.23T + 27T^{2} \) |
| 5 | \( 1 + 8.08T + 125T^{2} \) |
| 11 | \( 1 - 14.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 44.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 2.31T + 6.85e3T^{2} \) |
| 23 | \( 1 + 169.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 19.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 80.6T + 5.06e4T^{2} \) |
| 43 | \( 1 - 313.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 441.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 294.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 41.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 202.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 548.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 803.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 602.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 998.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 401.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 329.T + 9.12e5T^{2} \) |
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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
−9.021782736572099667396960816470, −8.059829029544600623435641909142, −7.67629346533977916477502885870, −6.83727865071495059809188457854, −5.57599652164292456473151563806, −4.20102843031574698839769217378, −3.66505547646730917909805257023, −2.72777869271192783244092498714, −1.72648418346976973473145186568, 0,
1.72648418346976973473145186568, 2.72777869271192783244092498714, 3.66505547646730917909805257023, 4.20102843031574698839769217378, 5.57599652164292456473151563806, 6.83727865071495059809188457854, 7.67629346533977916477502885870, 8.059829029544600623435641909142, 9.021782736572099667396960816470
