| L(s) = 1 | − 2.96·5-s − 3.35·7-s + 1.61·11-s − 13-s − 2·17-s − 3.35·19-s − 6.70·23-s + 3.77·25-s + 2·29-s − 6.57·31-s + 9.92·35-s − 7.92·37-s − 6.96·41-s − 0.775·43-s + 2.38·47-s + 4.22·49-s + 11.9·53-s − 4.77·55-s − 0.312·59-s − 14.6·61-s + 2.96·65-s + 8.12·67-s − 4.31·71-s + 0.0752·73-s − 5.40·77-s − 12·79-s + 8.31·83-s + ⋯ |
| L(s) = 1 | − 1.32·5-s − 1.26·7-s + 0.486·11-s − 0.277·13-s − 0.485·17-s − 0.768·19-s − 1.39·23-s + 0.755·25-s + 0.371·29-s − 1.18·31-s + 1.67·35-s − 1.30·37-s − 1.08·41-s − 0.118·43-s + 0.348·47-s + 0.603·49-s + 1.63·53-s − 0.643·55-s − 0.0407·59-s − 1.87·61-s + 0.367·65-s + 0.992·67-s − 0.511·71-s + 0.00880·73-s − 0.615·77-s − 1.35·79-s + 0.912·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2931874502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2931874502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2.96T + 5T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 6.57T + 31T^{2} \) |
| 37 | \( 1 + 7.92T + 37T^{2} \) |
| 41 | \( 1 + 6.96T + 41T^{2} \) |
| 43 | \( 1 + 0.775T + 43T^{2} \) |
| 47 | \( 1 - 2.38T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 0.312T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 - 8.12T + 67T^{2} \) |
| 71 | \( 1 + 4.31T + 71T^{2} \) |
| 73 | \( 1 - 0.0752T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 8.31T + 83T^{2} \) |
| 89 | \( 1 + 8.88T + 89T^{2} \) |
| 97 | \( 1 + 7.92T + 97T^{2} \) |
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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
−7.80375065502201123208548797731, −7.17718063645854889642127666526, −6.59994525584811427264891608365, −5.96594033839834943799422311517, −4.97289227468471342952572493082, −3.97174959070975493138730621444, −3.78934915884474282783729315906, −2.87615873775432490134393290629, −1.82087593807600260149522320580, −0.25817302538418022391279236360,
0.25817302538418022391279236360, 1.82087593807600260149522320580, 2.87615873775432490134393290629, 3.78934915884474282783729315906, 3.97174959070975493138730621444, 4.97289227468471342952572493082, 5.96594033839834943799422311517, 6.59994525584811427264891608365, 7.17718063645854889642127666526, 7.80375065502201123208548797731
