| L(s) = 1 | − 3·3-s + 0.239·5-s + 9·9-s − 8.09·11-s + 43.7·13-s − 0.717·15-s − 60.8·17-s + 98.8·19-s + 213.·23-s − 124.·25-s − 27·27-s + 110.·29-s + 80.0·31-s + 24.2·33-s − 2.88·37-s − 131.·39-s − 242.·41-s − 367.·43-s + 2.15·45-s − 89.3·47-s + 182.·51-s + 11.4·53-s − 1.93·55-s − 296.·57-s − 400.·59-s + 480.·61-s + 10.4·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.0214·5-s + 0.333·9-s − 0.221·11-s + 0.933·13-s − 0.0123·15-s − 0.867·17-s + 1.19·19-s + 1.93·23-s − 0.999·25-s − 0.192·27-s + 0.706·29-s + 0.463·31-s + 0.128·33-s − 0.0127·37-s − 0.539·39-s − 0.922·41-s − 1.30·43-s + 0.00713·45-s − 0.277·47-s + 0.500·51-s + 0.0295·53-s − 0.00475·55-s − 0.689·57-s − 0.882·59-s + 1.00·61-s + 0.0199·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.821592926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.821592926\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.239T + 125T^{2} \) |
| 11 | \( 1 + 8.09T + 1.33e3T^{2} \) |
| 13 | \( 1 - 43.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 98.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 213.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 80.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 2.88T + 5.06e4T^{2} \) |
| 41 | \( 1 + 242.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 367.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 89.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 11.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 400.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 240.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 978.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 622.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 545.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 845.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 203.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
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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
−8.632723139839631631230113505209, −7.87722000313051106597442393301, −6.88639118129966483453241280516, −6.42740008206991177415204234998, −5.37600400758288517992210065297, −4.85794536437771203177572405602, −3.76408257438773016348943870499, −2.89228948352240240050630762053, −1.59548246618918801868598923716, −0.64897443485950431254260398842,
0.64897443485950431254260398842, 1.59548246618918801868598923716, 2.89228948352240240050630762053, 3.76408257438773016348943870499, 4.85794536437771203177572405602, 5.37600400758288517992210065297, 6.42740008206991177415204234998, 6.88639118129966483453241280516, 7.87722000313051106597442393301, 8.632723139839631631230113505209
