| L(s) = 1 | − 3.53·2-s + 7.82·3-s + 4.46·4-s + 2.07·5-s − 27.6·6-s + 12.4·8-s + 34.1·9-s − 7.33·10-s + 49.1·11-s + 34.9·12-s + 44.8·13-s + 16.2·15-s − 79.7·16-s − 26.5·17-s − 120.·18-s − 77.7·19-s + 9.28·20-s − 173.·22-s + 55.7·23-s + 97.5·24-s − 120.·25-s − 158.·26-s + 56.2·27-s + 121.·29-s − 57.3·30-s − 305.·31-s + 181.·32-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 1.50·3-s + 0.558·4-s + 0.185·5-s − 1.87·6-s + 0.551·8-s + 1.26·9-s − 0.231·10-s + 1.34·11-s + 0.840·12-s + 0.956·13-s + 0.279·15-s − 1.24·16-s − 0.378·17-s − 1.58·18-s − 0.938·19-s + 0.103·20-s − 1.68·22-s + 0.505·23-s + 0.829·24-s − 0.965·25-s − 1.19·26-s + 0.400·27-s + 0.777·29-s − 0.349·30-s − 1.77·31-s + 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.202885298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.202885298\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 3.53T + 8T^{2} \) |
| 3 | \( 1 - 7.82T + 27T^{2} \) |
| 5 | \( 1 - 2.07T + 125T^{2} \) |
| 11 | \( 1 - 49.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 26.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 77.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 55.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 305.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 77.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 248.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 147.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 141.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 424.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 179.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 674.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 237.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 495.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 24.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.66e3T + 9.12e5T^{2} \) |
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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
−15.07934209573163466305004421863, −14.06138937785742708015713732426, −13.13666103958812530191013531636, −11.20257819491876418713911258748, −9.785279242392445464368840220259, −8.914017700696482534327261248755, −8.266133180865503445888015742762, −6.80570825296225348670572440666, −3.88051164760932429373629828840, −1.75457033817401880998895302335,
1.75457033817401880998895302335, 3.88051164760932429373629828840, 6.80570825296225348670572440666, 8.266133180865503445888015742762, 8.914017700696482534327261248755, 9.785279242392445464368840220259, 11.20257819491876418713911258748, 13.13666103958812530191013531636, 14.06138937785742708015713732426, 15.07934209573163466305004421863
