| L(s) = 1 | − 5.40·3-s − 9.23·5-s + 2.17·7-s + 2.26·9-s + 51.8·11-s + 10.4·13-s + 49.9·15-s + 17·17-s + 122.·19-s − 11.7·21-s + 48.3·23-s − 39.6·25-s + 133.·27-s + 160.·29-s + 12.2·31-s − 280.·33-s − 20.0·35-s − 103.·37-s − 56.3·39-s + 113.·41-s + 76.1·43-s − 20.8·45-s − 289.·47-s − 338.·49-s − 91.9·51-s + 447.·53-s − 479.·55-s + ⋯ |
| L(s) = 1 | − 1.04·3-s − 0.826·5-s + 0.117·7-s + 0.0837·9-s + 1.42·11-s + 0.222·13-s + 0.860·15-s + 0.242·17-s + 1.47·19-s − 0.121·21-s + 0.438·23-s − 0.317·25-s + 0.953·27-s + 1.02·29-s + 0.0708·31-s − 1.48·33-s − 0.0968·35-s − 0.458·37-s − 0.231·39-s + 0.431·41-s + 0.270·43-s − 0.0692·45-s − 0.899·47-s − 0.986·49-s − 0.252·51-s + 1.15·53-s − 1.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.017014079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.017014079\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 3 | \( 1 + 5.40T + 27T^{2} \) |
| 5 | \( 1 + 9.23T + 125T^{2} \) |
| 7 | \( 1 - 2.17T + 343T^{2} \) |
| 11 | \( 1 - 51.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.4T + 2.19e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 48.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 12.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 103.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 76.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 289.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 447.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 480.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 56.4T + 3.00e5T^{2} \) |
| 71 | \( 1 - 332.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 35.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 471.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 559.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 869.T + 9.12e5T^{2} \) |
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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
−12.25430993092420277448932791047, −11.73883290716476839758723291536, −11.07067795716351351913611778234, −9.725385231543213705327323261849, −8.485503171190854551554052479357, −7.19262748790368397574980780668, −6.12951909398383682340471947779, −4.87883782316510487428177837708, −3.52381042601690238190309292789, −0.923020105088074693502528918062,
0.923020105088074693502528918062, 3.52381042601690238190309292789, 4.87883782316510487428177837708, 6.12951909398383682340471947779, 7.19262748790368397574980780668, 8.485503171190854551554052479357, 9.725385231543213705327323261849, 11.07067795716351351913611778234, 11.73883290716476839758723291536, 12.25430993092420277448932791047
