| L(s) = 1 | + 2.31·2-s − 0.519·3-s + 3.36·4-s + 5-s − 1.20·6-s + 2·7-s + 3.16·8-s − 2.73·9-s + 2.31·10-s − 1.74·12-s + 13-s + 4.63·14-s − 0.519·15-s + 0.593·16-s + 3.43·17-s − 6.32·18-s − 2.63·19-s + 3.36·20-s − 1.03·21-s + 6.43·23-s − 1.64·24-s + 25-s + 2.31·26-s + 2.97·27-s + 6.73·28-s + 9.22·29-s − 1.20·30-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 0.299·3-s + 1.68·4-s + 0.447·5-s − 0.491·6-s + 0.755·7-s + 1.11·8-s − 0.910·9-s + 0.732·10-s − 0.504·12-s + 0.277·13-s + 1.23·14-s − 0.134·15-s + 0.148·16-s + 0.834·17-s − 1.49·18-s − 0.603·19-s + 0.752·20-s − 0.226·21-s + 1.34·23-s − 0.335·24-s + 0.200·25-s + 0.454·26-s + 0.572·27-s + 1.27·28-s + 1.71·29-s − 0.219·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7865 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.988644141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.988644141\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.31T + 2T^{2} \) |
| 3 | \( 1 + 0.519T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 17 | \( 1 - 3.43T + 17T^{2} \) |
| 19 | \( 1 + 2.63T + 19T^{2} \) |
| 23 | \( 1 - 6.43T + 23T^{2} \) |
| 29 | \( 1 - 9.22T + 29T^{2} \) |
| 31 | \( 1 - 4.84T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 + 2.84T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 4.57T + 53T^{2} \) |
| 59 | \( 1 + 1.78T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 - 4.40T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−7.60390769307827636256974707083, −6.81304371354469923971238770437, −6.16556524860153782475095181839, −5.67952185704732901435611267210, −4.88192918617937146528386140062, −4.67850371703265775881194305884, −3.50583393761436298341281587019, −2.92070868146695278613125627450, −2.13563494240061160160820528758, −0.993178432515267622774235963937,
0.993178432515267622774235963937, 2.13563494240061160160820528758, 2.92070868146695278613125627450, 3.50583393761436298341281587019, 4.67850371703265775881194305884, 4.88192918617937146528386140062, 5.67952185704732901435611267210, 6.16556524860153782475095181839, 6.81304371354469923971238770437, 7.60390769307827636256974707083
