| L(s) = 1 | − 3·3-s + 5.46·5-s − 12.3·7-s + 9·9-s − 6.78·11-s + 13·13-s − 16.3·15-s + 72.0·17-s − 99.2·19-s + 37.1·21-s − 120.·23-s − 95.1·25-s − 27·27-s + 185.·29-s + 85.4·31-s + 20.3·33-s − 67.7·35-s + 340.·37-s − 39·39-s − 427.·41-s + 64.9·43-s + 49.1·45-s + 39.2·47-s − 189.·49-s − 216.·51-s + 21.4·53-s − 37.0·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.488·5-s − 0.669·7-s + 0.333·9-s − 0.185·11-s + 0.277·13-s − 0.282·15-s + 1.02·17-s − 1.19·19-s + 0.386·21-s − 1.09·23-s − 0.761·25-s − 0.192·27-s + 1.18·29-s + 0.495·31-s + 0.107·33-s − 0.327·35-s + 1.51·37-s − 0.160·39-s − 1.62·41-s + 0.230·43-s + 0.162·45-s + 0.121·47-s − 0.552·49-s − 0.593·51-s + 0.0555·53-s − 0.0908·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.410745023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410745023\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 5.46T + 125T^{2} \) |
| 7 | \( 1 + 12.3T + 343T^{2} \) |
| 11 | \( 1 + 6.78T + 1.33e3T^{2} \) |
| 17 | \( 1 - 72.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 99.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 185.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 85.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 340.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 427.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 64.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 39.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 21.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 62.4T + 2.05e5T^{2} \) |
| 61 | \( 1 - 423.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 451.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 335.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 398.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 865.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 641.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−8.479360228838093779193916470292, −7.88335187427264595991189219480, −6.79590162055459088814552887895, −6.18098324769395376523947301197, −5.67712785077041955867717631772, −4.64263068239161157146201960069, −3.79960137742305090801049765714, −2.75160647357195239929332145033, −1.71088368964779093207535195851, −0.53627557505848555798188301609,
0.53627557505848555798188301609, 1.71088368964779093207535195851, 2.75160647357195239929332145033, 3.79960137742305090801049765714, 4.64263068239161157146201960069, 5.67712785077041955867717631772, 6.18098324769395376523947301197, 6.79590162055459088814552887895, 7.88335187427264595991189219480, 8.479360228838093779193916470292
