| L(s) = 1 | + 4.90·2-s + 0.777·3-s + 16.1·4-s + 3.81·6-s + 30.3·7-s + 39.7·8-s − 26.3·9-s + 22.4·11-s + 12.5·12-s + 62.0·13-s + 148.·14-s + 66.4·16-s + 72.1·17-s − 129.·18-s − 25.0·19-s + 23.5·21-s + 110.·22-s − 103.·23-s + 30.9·24-s + 304.·26-s − 41.5·27-s + 488.·28-s − 234.·29-s + 198.·31-s + 8.17·32-s + 17.4·33-s + 354.·34-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 0.149·3-s + 2.01·4-s + 0.259·6-s + 1.63·7-s + 1.75·8-s − 0.977·9-s + 0.614·11-s + 0.301·12-s + 1.32·13-s + 2.84·14-s + 1.03·16-s + 1.02·17-s − 1.69·18-s − 0.302·19-s + 0.245·21-s + 1.06·22-s − 0.935·23-s + 0.263·24-s + 2.29·26-s − 0.296·27-s + 3.29·28-s − 1.50·29-s + 1.15·31-s + 0.0451·32-s + 0.0920·33-s + 1.78·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.455390720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.455390720\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 47 | \( 1 + 47T \) |
| good | 2 | \( 1 - 4.90T + 8T^{2} \) |
| 3 | \( 1 - 0.777T + 27T^{2} \) |
| 7 | \( 1 - 30.3T + 343T^{2} \) |
| 11 | \( 1 - 22.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 25.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 198.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 203.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 111.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 499.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 562.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 548.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 760.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 668.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 975.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 698.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 451.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 390.T + 9.12e5T^{2} \) |
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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
−9.335544303255047228497124392522, −8.215761791127584658506673134632, −7.81489064801472464435350272364, −6.43829939032793818162982488721, −5.81889215223919628430680848493, −5.11281542767261898123130230537, −4.15357313851768148657754762343, −3.49711640114080504664406331807, −2.30536584016938211582356448315, −1.32077201698659161111203176339,
1.32077201698659161111203176339, 2.30536584016938211582356448315, 3.49711640114080504664406331807, 4.15357313851768148657754762343, 5.11281542767261898123130230537, 5.81889215223919628430680848493, 6.43829939032793818162982488721, 7.81489064801472464435350272364, 8.215761791127584658506673134632, 9.335544303255047228497124392522
