| L(s) = 1 | − 5·5-s + 3.85·7-s − 42.2·11-s + 4.96·13-s − 25.8·17-s − 28.9·19-s + 191.·23-s + 25·25-s + 287.·29-s − 52.6·31-s − 19.2·35-s − 225.·37-s + 73.9·41-s + 275.·43-s + 192.·47-s − 328.·49-s + 275.·53-s + 211.·55-s − 497.·59-s + 44.0·61-s − 24.8·65-s − 761.·67-s − 264.·71-s + 728.·73-s − 163.·77-s − 664.·79-s − 1.49e3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.208·7-s − 1.15·11-s + 0.105·13-s − 0.369·17-s − 0.350·19-s + 1.73·23-s + 0.200·25-s + 1.84·29-s − 0.305·31-s − 0.0931·35-s − 1.00·37-s + 0.281·41-s + 0.976·43-s + 0.596·47-s − 0.956·49-s + 0.713·53-s + 0.518·55-s − 1.09·59-s + 0.0924·61-s − 0.0473·65-s − 1.38·67-s − 0.441·71-s + 1.16·73-s − 0.241·77-s − 0.946·79-s − 1.97·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| good | 7 | \( 1 - 3.85T + 343T^{2} \) |
| 11 | \( 1 + 42.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.96T + 2.19e3T^{2} \) |
| 17 | \( 1 + 25.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 191.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 287.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 52.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 225.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 73.9T + 6.89e4T^{2} \) |
| 43 | \( 1 - 275.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 192.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 275.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 497.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 44.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 761.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 264.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 728.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 664.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.49e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 106.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 924.T + 9.12e5T^{2} \) |
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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
−8.440526914230067689464299867197, −7.51380784336560094511164234462, −6.93942687212484658800974692089, −5.93411563624558555385146452136, −4.99404661194005712066647994286, −4.43220851127969011432205508804, −3.20989566167867321409531826901, −2.49957748022951357093666982955, −1.14945861957921636046542169205, 0,
1.14945861957921636046542169205, 2.49957748022951357093666982955, 3.20989566167867321409531826901, 4.43220851127969011432205508804, 4.99404661194005712066647994286, 5.93411563624558555385146452136, 6.93942687212484658800974692089, 7.51380784336560094511164234462, 8.440526914230067689464299867197
