| L(s) = 1 | − 3-s − 4.39·5-s + 7-s + 9-s − 3.51·11-s − 5.51·13-s + 4.39·15-s + 3.11·17-s + 5.05·19-s − 21-s − 23-s + 14.3·25-s − 27-s − 6.66·29-s − 1.11·31-s + 3.51·33-s − 4.39·35-s − 6.66·37-s + 5.51·39-s − 10.6·41-s − 6.81·43-s − 4.39·45-s − 3.43·47-s + 49-s − 3.11·51-s − 9.54·53-s + 15.4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.96·5-s + 0.377·7-s + 0.333·9-s − 1.06·11-s − 1.53·13-s + 1.13·15-s + 0.756·17-s + 1.16·19-s − 0.218·21-s − 0.208·23-s + 2.86·25-s − 0.192·27-s − 1.23·29-s − 0.201·31-s + 0.612·33-s − 0.743·35-s − 1.09·37-s + 0.883·39-s − 1.65·41-s − 1.03·43-s − 0.655·45-s − 0.500·47-s + 0.142·49-s − 0.436·51-s − 1.31·53-s + 2.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2156060916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2156060916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 4.39T + 5T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 + 5.51T + 13T^{2} \) |
| 17 | \( 1 - 3.11T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 29 | \( 1 + 6.66T + 29T^{2} \) |
| 31 | \( 1 + 1.11T + 31T^{2} \) |
| 37 | \( 1 + 6.66T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 6.81T + 43T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 + 9.54T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 6.11T + 67T^{2} \) |
| 71 | \( 1 + 8.02T + 71T^{2} \) |
| 73 | \( 1 + 2.95T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 9.91T + 83T^{2} \) |
| 89 | \( 1 + 4.31T + 89T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−7.74664409481070622665838629910, −7.35536323506574317800206427715, −6.77945367299144123028475976879, −5.38787621791960752599005121202, −5.10547373204835074370427244281, −4.45127833330681886957508028421, −3.47647705216944626669649619281, −2.98769471939414362175961692549, −1.61536622477408555232535809760, −0.23653248422912039456199668202,
0.23653248422912039456199668202, 1.61536622477408555232535809760, 2.98769471939414362175961692549, 3.47647705216944626669649619281, 4.45127833330681886957508028421, 5.10547373204835074370427244281, 5.38787621791960752599005121202, 6.77945367299144123028475976879, 7.35536323506574317800206427715, 7.74664409481070622665838629910
