| L(s) = 1 | + 1.07e12·2-s − 9.45e18·3-s + 5.49e23·4-s − 5.27e26·5-s − 1.01e31·6-s − 3.45e33·7-s − 5.89e34·8-s + 4.00e37·9-s − 5.66e38·10-s + 2.33e41·11-s − 5.19e42·12-s − 5.41e43·13-s − 3.70e45·14-s + 4.98e45·15-s − 3.95e47·16-s + 5.59e48·17-s + 4.30e49·18-s − 2.84e48·19-s − 2.90e50·20-s + 3.26e52·21-s + 2.50e53·22-s − 7.81e52·23-s + 5.57e53·24-s − 1.62e55·25-s − 5.82e55·26-s + 8.68e55·27-s − 1.89e57·28-s + ⋯ |
| L(s) = 1 | + 1.38·2-s − 1.34·3-s + 0.909·4-s − 0.129·5-s − 1.86·6-s − 1.43·7-s − 0.125·8-s + 0.813·9-s − 0.179·10-s + 1.71·11-s − 1.22·12-s − 0.541·13-s − 1.98·14-s + 0.174·15-s − 1.08·16-s + 1.39·17-s + 1.12·18-s − 0.00877·19-s − 0.117·20-s + 1.93·21-s + 2.36·22-s − 0.127·23-s + 0.168·24-s − 0.983·25-s − 0.747·26-s + 0.251·27-s − 1.30·28-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
| \(L(40)\) |
\(\approx\) |
\(1.972707396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.972707396\) |
| \(L(\frac{81}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.07e12T + 6.04e23T^{2} \) |
| 3 | \( 1 + 9.45e18T + 4.92e37T^{2} \) |
| 5 | \( 1 + 5.27e26T + 1.65e55T^{2} \) |
| 7 | \( 1 + 3.45e33T + 5.79e66T^{2} \) |
| 11 | \( 1 - 2.33e41T + 1.86e82T^{2} \) |
| 13 | \( 1 + 5.41e43T + 1.00e88T^{2} \) |
| 17 | \( 1 - 5.59e48T + 1.60e97T^{2} \) |
| 19 | \( 1 + 2.84e48T + 1.05e101T^{2} \) |
| 23 | \( 1 + 7.81e52T + 3.77e107T^{2} \) |
| 29 | \( 1 - 9.18e57T + 3.38e115T^{2} \) |
| 31 | \( 1 - 1.25e59T + 6.57e117T^{2} \) |
| 37 | \( 1 - 3.95e61T + 7.72e123T^{2} \) |
| 41 | \( 1 - 5.78e62T + 2.56e127T^{2} \) |
| 43 | \( 1 - 2.81e64T + 1.10e129T^{2} \) |
| 47 | \( 1 + 2.26e65T + 1.24e132T^{2} \) |
| 53 | \( 1 + 1.18e68T + 1.65e136T^{2} \) |
| 59 | \( 1 + 1.26e70T + 7.89e139T^{2} \) |
| 61 | \( 1 - 2.75e70T + 1.09e141T^{2} \) |
| 67 | \( 1 - 6.88e71T + 1.81e144T^{2} \) |
| 71 | \( 1 - 2.19e72T + 1.77e146T^{2} \) |
| 73 | \( 1 - 1.79e72T + 1.59e147T^{2} \) |
| 79 | \( 1 + 1.14e75T + 8.17e149T^{2} \) |
| 83 | \( 1 - 6.74e75T + 4.04e151T^{2} \) |
| 89 | \( 1 - 3.78e76T + 1.00e154T^{2} \) |
| 97 | \( 1 + 2.34e78T + 9.01e156T^{2} \) |
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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
−15.94332465364546704083176052310, −14.14224448913184316559226436494, −12.39973495273673063464461914084, −11.82448633633861174051484207112, −9.765250113796691346170925918342, −6.58509434788769017196571620612, −5.94146884513462250177598342614, −4.43494186680390215756685092359, −3.17174258368265871893196000686, −0.72084184862080328275728459102,
0.72084184862080328275728459102, 3.17174258368265871893196000686, 4.43494186680390215756685092359, 5.94146884513462250177598342614, 6.58509434788769017196571620612, 9.765250113796691346170925918342, 11.82448633633861174051484207112, 12.39973495273673063464461914084, 14.14224448913184316559226436494, 15.94332465364546704083176052310
