| L(s) = 1 | + 5.97·3-s − 12.4·5-s + 28.6·7-s + 8.64·9-s + 18.2·11-s + 37.9·13-s − 74.5·15-s − 3.95·17-s + 36.8·19-s + 171.·21-s − 42.7·23-s + 30.9·25-s − 109.·27-s − 29·29-s + 160.·31-s + 108.·33-s − 358.·35-s + 313.·37-s + 226.·39-s + 496.·41-s + 139.·43-s − 107.·45-s + 417.·47-s + 479.·49-s − 23.5·51-s − 137.·53-s − 227.·55-s + ⋯ |
| L(s) = 1 | + 1.14·3-s − 1.11·5-s + 1.54·7-s + 0.320·9-s + 0.499·11-s + 0.810·13-s − 1.28·15-s − 0.0563·17-s + 0.444·19-s + 1.77·21-s − 0.387·23-s + 0.247·25-s − 0.781·27-s − 0.185·29-s + 0.931·31-s + 0.573·33-s − 1.72·35-s + 1.39·37-s + 0.931·39-s + 1.89·41-s + 0.493·43-s − 0.357·45-s + 1.29·47-s + 1.39·49-s − 0.0647·51-s − 0.355·53-s − 0.557·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.983134609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.983134609\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 3 | \( 1 - 5.97T + 27T^{2} \) |
| 5 | \( 1 + 12.4T + 125T^{2} \) |
| 7 | \( 1 - 28.6T + 343T^{2} \) |
| 11 | \( 1 - 18.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.95T + 4.91e3T^{2} \) |
| 19 | \( 1 - 36.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 42.7T + 1.21e4T^{2} \) |
| 31 | \( 1 - 160.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 313.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 496.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 139.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 417.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 137.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 190.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 161.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 125.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 165.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 938.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 505.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 769.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.33e3T + 9.12e5T^{2} \) |
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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
−10.88983690444926483831893674733, −9.430615798346768330370731753258, −8.605518140922602634274396435734, −7.943535420446054647030105970367, −7.49694816305070879236507261126, −5.90842592169912868629374147920, −4.44932930965377188542457333597, −3.82247667414419786027880566385, −2.51894970812335085969247349872, −1.13264370554835181153374287161,
1.13264370554835181153374287161, 2.51894970812335085969247349872, 3.82247667414419786027880566385, 4.44932930965377188542457333597, 5.90842592169912868629374147920, 7.49694816305070879236507261126, 7.943535420446054647030105970367, 8.605518140922602634274396435734, 9.430615798346768330370731753258, 10.88983690444926483831893674733
