| L(s) = 1 | + 4·2-s + 8.15·3-s + 16·4-s + 86.4·5-s + 32.6·6-s + 72.3·7-s + 64·8-s − 176.·9-s + 345.·10-s − 547.·11-s + 130.·12-s + 630.·13-s + 289.·14-s + 704.·15-s + 256·16-s − 202.·17-s − 706.·18-s + 2.31e3·19-s + 1.38e3·20-s + 589.·21-s − 2.19e3·22-s − 1.34e3·23-s + 521.·24-s + 4.35e3·25-s + 2.52e3·26-s − 3.41e3·27-s + 1.15e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.522·3-s + 0.5·4-s + 1.54·5-s + 0.369·6-s + 0.558·7-s + 0.353·8-s − 0.726·9-s + 1.09·10-s − 1.36·11-s + 0.261·12-s + 1.03·13-s + 0.394·14-s + 0.808·15-s + 0.250·16-s − 0.169·17-s − 0.513·18-s + 1.47·19-s + 0.773·20-s + 0.291·21-s − 0.964·22-s − 0.531·23-s + 0.184·24-s + 1.39·25-s + 0.731·26-s − 0.902·27-s + 0.279·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.859668657\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.859668657\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 37 | \( 1 + 1.36e3T \) |
| good | 3 | \( 1 - 8.15T + 243T^{2} \) |
| 5 | \( 1 - 86.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 72.3T + 1.68e4T^{2} \) |
| 11 | \( 1 + 547.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 630.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 202.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.34e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 634.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.38e3T + 2.86e7T^{2} \) |
| 41 | \( 1 + 496.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.23e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 868.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.31e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.09e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.55e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.63e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.95e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.38e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.42e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.48e5T + 8.58e9T^{2} \) |
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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
−13.76001444883983610228427645316, −12.95135621500068002567959116890, −11.36414943357461162069313753820, −10.31469474441338473312950452812, −9.035727352536272875301661010992, −7.74350485746252364300726240488, −5.96619079079016509623036066638, −5.21781172563474506161129652273, −3.09014218817677445372765279395, −1.83851338775316483428800257983,
1.83851338775316483428800257983, 3.09014218817677445372765279395, 5.21781172563474506161129652273, 5.96619079079016509623036066638, 7.74350485746252364300726240488, 9.035727352536272875301661010992, 10.31469474441338473312950452812, 11.36414943357461162069313753820, 12.95135621500068002567959116890, 13.76001444883983610228427645316
