| L(s) = 1 | − 4.75·3-s − 10.5·5-s − 30.4·7-s − 4.38·9-s − 35.9·11-s + 84.5·13-s + 50.2·15-s + 46.5·17-s + 19·19-s + 144.·21-s − 21.5·23-s − 13.1·25-s + 149.·27-s + 241.·29-s + 143.·31-s + 171.·33-s + 321.·35-s − 57.8·37-s − 402.·39-s − 305.·41-s + 314.·43-s + 46.4·45-s − 228.·47-s + 583.·49-s − 221.·51-s + 85.6·53-s + 380.·55-s + ⋯ |
| L(s) = 1 | − 0.915·3-s − 0.945·5-s − 1.64·7-s − 0.162·9-s − 0.985·11-s + 1.80·13-s + 0.865·15-s + 0.664·17-s + 0.229·19-s + 1.50·21-s − 0.195·23-s − 0.105·25-s + 1.06·27-s + 1.54·29-s + 0.831·31-s + 0.902·33-s + 1.55·35-s − 0.257·37-s − 1.65·39-s − 1.16·41-s + 1.11·43-s + 0.153·45-s − 0.709·47-s + 1.70·49-s − 0.608·51-s + 0.222·53-s + 0.932·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 4.75T + 27T^{2} \) |
| 5 | \( 1 + 10.5T + 125T^{2} \) |
| 7 | \( 1 + 30.4T + 343T^{2} \) |
| 11 | \( 1 + 35.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 84.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 46.5T + 4.91e3T^{2} \) |
| 23 | \( 1 + 21.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 143.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 57.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 305.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 314.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 228.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 85.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 52.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 133.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.07e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 245.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 501.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 730.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 915.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 374.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 440.T + 9.12e5T^{2} \) |
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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
−8.814558135604949058424067669902, −8.190825895980169569167556667950, −7.17079121655013746827869269413, −6.20217311491306754268157675967, −5.87813329181579674941305608869, −4.66652315215209614416450324358, −3.52335657323917813999371916021, −2.96123142424159914835232192407, −0.885535244294897699887810656034, 0,
0.885535244294897699887810656034, 2.96123142424159914835232192407, 3.52335657323917813999371916021, 4.66652315215209614416450324358, 5.87813329181579674941305608869, 6.20217311491306754268157675967, 7.17079121655013746827869269413, 8.190825895980169569167556667950, 8.814558135604949058424067669902
