| L(s) = 1 | + 3-s − 2.61·5-s − 0.236·7-s − 2·9-s + 3·11-s − 13-s − 2.61·15-s − 5.09·17-s − 3.09·19-s − 0.236·21-s + 8.23·23-s + 1.85·25-s − 5·27-s − 3.09·29-s − 2·31-s + 3·33-s + 0.618·35-s + 3.85·37-s − 39-s + 7.32·41-s + 8.56·43-s + 5.23·45-s + 5.61·47-s − 6.94·49-s − 5.09·51-s + 10.1·53-s − 7.85·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.17·5-s − 0.0892·7-s − 0.666·9-s + 0.904·11-s − 0.277·13-s − 0.675·15-s − 1.23·17-s − 0.708·19-s − 0.0515·21-s + 1.71·23-s + 0.370·25-s − 0.962·27-s − 0.573·29-s − 0.359·31-s + 0.522·33-s + 0.104·35-s + 0.633·37-s − 0.160·39-s + 1.14·41-s + 1.30·43-s + 0.780·45-s + 0.819·47-s − 0.992·49-s − 0.712·51-s + 1.39·53-s − 1.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.420485990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420485990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 + 0.236T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 3.09T + 19T^{2} \) |
| 23 | \( 1 - 8.23T + 23T^{2} \) |
| 29 | \( 1 + 3.09T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 3.85T + 37T^{2} \) |
| 41 | \( 1 - 7.32T + 41T^{2} \) |
| 43 | \( 1 - 8.56T + 43T^{2} \) |
| 47 | \( 1 - 5.61T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 0.236T + 67T^{2} \) |
| 71 | \( 1 + 3.38T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 0.527T + 79T^{2} \) |
| 83 | \( 1 - 2.90T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 1.29T + 97T^{2} \) |
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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
−8.555050083843348281132330202088, −7.72602216840795671101124489466, −7.12580761186894752742000261489, −6.39704335069644747212942232215, −5.44048090838696214310696206300, −4.34290831727850655050541249369, −3.93414423620168065617763477034, −2.99038573947497763844827952304, −2.17891750787154942610401321277, −0.64229058928860339555013312861,
0.64229058928860339555013312861, 2.17891750787154942610401321277, 2.99038573947497763844827952304, 3.93414423620168065617763477034, 4.34290831727850655050541249369, 5.44048090838696214310696206300, 6.39704335069644747212942232215, 7.12580761186894752742000261489, 7.72602216840795671101124489466, 8.555050083843348281132330202088
