| 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
| \(L(2)\) |
\(\approx\) |
\(0.4031607051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4031607051\) |
| \(L(\frac{5}{2})\) |
|
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
−15.68669713416454609552086092983, −13.99294785824218850681192144988, −12.16931313324275024383024928713, −11.50655608047075882804097382656, −10.26760446400522511369056728052, −9.326144726423956130721730438880, −7.64099400864568382924892399128, −6.39820875894027246898174258689, −4.70001391395699142531083415200, −0.899811362693603773500375190422,
0.899811362693603773500375190422, 4.70001391395699142531083415200, 6.39820875894027246898174258689, 7.64099400864568382924892399128, 9.326144726423956130721730438880, 10.26760446400522511369056728052, 11.50655608047075882804097382656, 12.16931313324275024383024928713, 13.99294785824218850681192144988, 15.68669713416454609552086092983
