| L(s) = 1 | − 21.2·5-s − 29.2·7-s + 11-s + 52.9·13-s − 96.9·17-s + 126.·19-s + 22.9·23-s + 325.·25-s + 133.·29-s − 101.·31-s + 620.·35-s − 105.·37-s + 16.3·41-s − 201.·43-s + 251.·47-s + 511.·49-s + 148.·53-s − 21.2·55-s − 73.6·59-s + 607.·61-s − 1.12e3·65-s + 761.·67-s − 701.·71-s − 287·73-s − 29.2·77-s + 128.·79-s + 160.·83-s + ⋯ |
| L(s) = 1 | − 1.89·5-s − 1.57·7-s + 0.0274·11-s + 1.12·13-s − 1.38·17-s + 1.52·19-s + 0.207·23-s + 2.60·25-s + 0.855·29-s − 0.589·31-s + 2.99·35-s − 0.466·37-s + 0.0622·41-s − 0.713·43-s + 0.781·47-s + 1.49·49-s + 0.383·53-s − 0.0520·55-s − 0.162·59-s + 1.27·61-s − 2.14·65-s + 1.38·67-s − 1.17·71-s − 0.460·73-s − 0.0432·77-s + 0.183·79-s + 0.212·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 21.2T + 125T^{2} \) |
| 7 | \( 1 + 29.2T + 343T^{2} \) |
| 11 | \( 1 - T + 1.33e3T^{2} \) |
| 13 | \( 1 - 52.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 96.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 126.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 22.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 101.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 105.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 16.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 201.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 251.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 148.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 73.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 607.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 761.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 701.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 287T + 3.89e5T^{2} \) |
| 79 | \( 1 - 128.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 160.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 430.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 31.1T + 9.12e5T^{2} \) |
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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
−8.657979658602724198801599336123, −7.71865936058435891845395438500, −6.93824572115817663177308935794, −6.45057268781794376156963981019, −5.21105601935098106047618011095, −4.07034567652829383277988155426, −3.56525411132870476683038887215, −2.82851116980328152995142854890, −0.902024348303270334487324684550, 0,
0.902024348303270334487324684550, 2.82851116980328152995142854890, 3.56525411132870476683038887215, 4.07034567652829383277988155426, 5.21105601935098106047618011095, 6.45057268781794376156963981019, 6.93824572115817663177308935794, 7.71865936058435891845395438500, 8.657979658602724198801599336123
