| L(s) = 1 | + 4.20·3-s − 15.0·5-s + 31.2·7-s − 9.35·9-s − 14.1·11-s − 60.2·13-s − 63.4·15-s + 124.·17-s + 14.8·19-s + 131.·21-s + 182.·23-s + 102.·25-s − 152.·27-s + 154.·29-s + 119.·31-s − 59.6·33-s − 471.·35-s − 37·37-s − 253.·39-s − 87.7·41-s + 101.·43-s + 141.·45-s + 264.·47-s + 633.·49-s + 521.·51-s + 186.·53-s + 214.·55-s + ⋯ |
| L(s) = 1 | + 0.808·3-s − 1.35·5-s + 1.68·7-s − 0.346·9-s − 0.389·11-s − 1.28·13-s − 1.09·15-s + 1.77·17-s + 0.179·19-s + 1.36·21-s + 1.65·23-s + 0.823·25-s − 1.08·27-s + 0.988·29-s + 0.692·31-s − 0.314·33-s − 2.27·35-s − 0.164·37-s − 1.03·39-s − 0.334·41-s + 0.358·43-s + 0.467·45-s + 0.822·47-s + 1.84·49-s + 1.43·51-s + 0.483·53-s + 0.525·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.333340686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.333340686\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + 37T \) |
| good | 3 | \( 1 - 4.20T + 27T^{2} \) |
| 5 | \( 1 + 15.0T + 125T^{2} \) |
| 7 | \( 1 - 31.2T + 343T^{2} \) |
| 11 | \( 1 + 14.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 182.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 119.T + 2.97e4T^{2} \) |
| 41 | \( 1 + 87.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 264.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 186.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 611.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 46.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 246.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 987.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 547.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.18e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 958.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 255.T + 9.12e5T^{2} \) |
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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
−10.36709833652951950589828943595, −9.182875426721430105114150294792, −8.159374770256046760460017359417, −7.900487836976344014440788254869, −7.20565678787749829754844363774, −5.30486757538173390813391325315, −4.70019524855337729800091602949, −3.46429305226242507102657277985, −2.50678477578336549929764486567, −0.904244507468375672496720800468,
0.904244507468375672496720800468, 2.50678477578336549929764486567, 3.46429305226242507102657277985, 4.70019524855337729800091602949, 5.30486757538173390813391325315, 7.20565678787749829754844363774, 7.900487836976344014440788254869, 8.159374770256046760460017359417, 9.182875426721430105114150294792, 10.36709833652951950589828943595
