| L(s) = 1 | − 0.726·3-s − 0.236·5-s − 3.80·7-s − 2.47·9-s − 5.42·11-s + 6.23·13-s + 0.171·15-s − 0.763·17-s + 1.45·19-s + 2.76·21-s − 2.35·23-s − 4.94·25-s + 3.97·27-s − 29-s + 3.07·31-s + 3.94·33-s + 0.898·35-s + 6.47·37-s − 4.53·39-s − 2.76·41-s + 11.5·43-s + 0.583·45-s + 4.53·47-s + 7.47·49-s + 0.555·51-s + 8.23·53-s + 1.28·55-s + ⋯ |
| L(s) = 1 | − 0.419·3-s − 0.105·5-s − 1.43·7-s − 0.824·9-s − 1.63·11-s + 1.72·13-s + 0.0442·15-s − 0.185·17-s + 0.333·19-s + 0.603·21-s − 0.490·23-s − 0.988·25-s + 0.765·27-s − 0.185·29-s + 0.552·31-s + 0.686·33-s + 0.151·35-s + 1.06·37-s − 0.725·39-s − 0.431·41-s + 1.76·43-s + 0.0869·45-s + 0.660·47-s + 1.06·49-s + 0.0777·51-s + 1.13·53-s + 0.172·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7872535603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7872535603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 0.726T + 3T^{2} \) |
| 5 | \( 1 + 0.236T + 5T^{2} \) |
| 7 | \( 1 + 3.80T + 7T^{2} \) |
| 11 | \( 1 + 5.42T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 - 1.45T + 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 + 2.76T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 4.53T + 47T^{2} \) |
| 53 | \( 1 - 8.23T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 + 2.90T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 9.23T + 79T^{2} \) |
| 83 | \( 1 - 0.555T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−9.294191124890118734158749876027, −8.398054416497937150866727995511, −7.77736314632055628006626523846, −6.67606889851466290907075179895, −5.87114789946794948394606307339, −5.59998386704035006482378808046, −4.18117188201973129740611297824, −3.26486204668993406807111071930, −2.48178921483843553845839886585, −0.58203824929804832487447260960,
0.58203824929804832487447260960, 2.48178921483843553845839886585, 3.26486204668993406807111071930, 4.18117188201973129740611297824, 5.59998386704035006482378808046, 5.87114789946794948394606307339, 6.67606889851466290907075179895, 7.77736314632055628006626523846, 8.398054416497937150866727995511, 9.294191124890118734158749876027
