| L(s) = 1 | − 8.42e11·2-s + 1.18e19·3-s + 1.05e23·4-s + 5.37e27·5-s − 9.97e30·6-s − 2.88e33·7-s + 4.20e35·8-s + 9.08e37·9-s − 4.52e39·10-s + 1.27e41·11-s + 1.24e42·12-s + 8.81e43·13-s + 2.42e45·14-s + 6.36e46·15-s − 4.18e47·16-s − 2.88e47·17-s − 7.65e49·18-s + 2.48e50·19-s + 5.66e50·20-s − 3.41e52·21-s − 1.07e53·22-s − 9.54e53·23-s + 4.97e54·24-s + 1.23e55·25-s − 7.43e55·26-s + 4.92e56·27-s − 3.04e56·28-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 1.68·3-s + 0.174·4-s + 1.32·5-s − 1.82·6-s − 1.19·7-s + 0.894·8-s + 1.84·9-s − 1.43·10-s + 0.935·11-s + 0.294·12-s + 0.880·13-s + 1.29·14-s + 2.22·15-s − 1.14·16-s − 0.0721·17-s − 1.99·18-s + 0.765·19-s + 0.230·20-s − 2.02·21-s − 1.01·22-s − 1.55·23-s + 1.50·24-s + 0.744·25-s − 0.954·26-s + 1.42·27-s − 0.209·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(80-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+79/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(40)\) |
\(\approx\) |
\(2.562982764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.562982764\) |
| \(L(\frac{81}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 8.42e11T + 6.04e23T^{2} \) |
| 3 | \( 1 - 1.18e19T + 4.92e37T^{2} \) |
| 5 | \( 1 - 5.37e27T + 1.65e55T^{2} \) |
| 7 | \( 1 + 2.88e33T + 5.79e66T^{2} \) |
| 11 | \( 1 - 1.27e41T + 1.86e82T^{2} \) |
| 13 | \( 1 - 8.81e43T + 1.00e88T^{2} \) |
| 17 | \( 1 + 2.88e47T + 1.60e97T^{2} \) |
| 19 | \( 1 - 2.48e50T + 1.05e101T^{2} \) |
| 23 | \( 1 + 9.54e53T + 3.77e107T^{2} \) |
| 29 | \( 1 - 2.62e57T + 3.38e115T^{2} \) |
| 31 | \( 1 - 5.35e58T + 6.57e117T^{2} \) |
| 37 | \( 1 - 1.24e62T + 7.72e123T^{2} \) |
| 41 | \( 1 - 4.32e62T + 2.56e127T^{2} \) |
| 43 | \( 1 + 2.59e64T + 1.10e129T^{2} \) |
| 47 | \( 1 - 1.81e65T + 1.24e132T^{2} \) |
| 53 | \( 1 + 1.64e68T + 1.65e136T^{2} \) |
| 59 | \( 1 - 1.72e70T + 7.89e139T^{2} \) |
| 61 | \( 1 - 3.55e70T + 1.09e141T^{2} \) |
| 67 | \( 1 - 1.15e71T + 1.81e144T^{2} \) |
| 71 | \( 1 - 1.64e73T + 1.77e146T^{2} \) |
| 73 | \( 1 + 2.56e72T + 1.59e147T^{2} \) |
| 79 | \( 1 - 1.45e75T + 8.17e149T^{2} \) |
| 83 | \( 1 - 5.44e75T + 4.04e151T^{2} \) |
| 89 | \( 1 - 1.18e76T + 1.00e154T^{2} \) |
| 97 | \( 1 + 2.81e78T + 9.01e156T^{2} \) |
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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
−16.18906363462435269500334942312, −14.06286354893766375040193102653, −13.31259098394529976760683078015, −9.844507010027583086402463229156, −9.469484718529697102187267081141, −8.252996609602681201486189189326, −6.48436546906874165710182175042, −3.74447914094962328498607883221, −2.27486611006901282937269064041, −1.13148549752498362450558244495,
1.13148549752498362450558244495, 2.27486611006901282937269064041, 3.74447914094962328498607883221, 6.48436546906874165710182175042, 8.252996609602681201486189189326, 9.469484718529697102187267081141, 9.844507010027583086402463229156, 13.31259098394529976760683078015, 14.06286354893766375040193102653, 16.18906363462435269500334942312
