| L(s) = 1 | − 5·5-s − 10.2·7-s − 5.26·11-s + 34.5·13-s + 74.8·17-s − 50.8·19-s − 81.4·23-s + 25·25-s + 152.·29-s + 93.9·31-s + 51.3·35-s − 167.·37-s + 18.6·41-s + 133.·43-s + 46.8·47-s − 237.·49-s − 105.·53-s + 26.3·55-s − 77.7·59-s − 48.6·61-s − 172.·65-s − 667.·67-s − 344.·71-s − 1.07e3·73-s + 54·77-s − 420.·79-s − 296.·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.554·7-s − 0.144·11-s + 0.737·13-s + 1.06·17-s − 0.614·19-s − 0.738·23-s + 0.200·25-s + 0.975·29-s + 0.544·31-s + 0.247·35-s − 0.743·37-s + 0.0709·41-s + 0.473·43-s + 0.145·47-s − 0.692·49-s − 0.272·53-s + 0.0645·55-s − 0.171·59-s − 0.102·61-s − 0.329·65-s − 1.21·67-s − 0.575·71-s − 1.72·73-s + 0.0799·77-s − 0.599·79-s − 0.392·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 10.2T + 343T^{2} \) |
| 11 | \( 1 + 5.26T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 74.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 81.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 152.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 93.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 18.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 133.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 46.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 105.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 77.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 48.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 667.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 344.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 420.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 296.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 950.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 135.T + 9.12e5T^{2} \) |
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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
−9.000480163694155093614389321247, −8.265733446289674862428613317393, −7.49730316443193557131636378955, −6.48037208318247311375719944914, −5.80302648800786435517421828719, −4.62036492467938615164697103065, −3.68268343406204405075373969307, −2.80727315302643448172379127722, −1.32283419650471605258791594628, 0,
1.32283419650471605258791594628, 2.80727315302643448172379127722, 3.68268343406204405075373969307, 4.62036492467938615164697103065, 5.80302648800786435517421828719, 6.48037208318247311375719944914, 7.49730316443193557131636378955, 8.265733446289674862428613317393, 9.000480163694155093614389321247
