| L(s) = 1 | + 5-s − 1.90·7-s + 5.48·11-s − 1.04·13-s + 6.74·17-s − 1.55·19-s + 23-s + 25-s + 3.38·29-s + 10.9·31-s − 1.90·35-s + 5.26·37-s − 6.09·41-s − 0.403·47-s − 3.36·49-s − 5.88·53-s + 5.48·55-s − 9.60·59-s − 7.09·61-s − 1.04·65-s + 13.7·67-s − 0.478·71-s + 2.40·73-s − 10.4·77-s − 4.24·79-s + 11.2·83-s + 6.74·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.720·7-s + 1.65·11-s − 0.291·13-s + 1.63·17-s − 0.355·19-s + 0.208·23-s + 0.200·25-s + 0.628·29-s + 1.96·31-s − 0.322·35-s + 0.865·37-s − 0.952·41-s − 0.0589·47-s − 0.480·49-s − 0.808·53-s + 0.738·55-s − 1.24·59-s − 0.908·61-s − 0.130·65-s + 1.68·67-s − 0.0568·71-s + 0.281·73-s − 1.19·77-s − 0.477·79-s + 1.23·83-s + 0.731·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.544145284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.544145284\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 - 5.48T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 - 6.74T + 17T^{2} \) |
| 19 | \( 1 + 1.55T + 19T^{2} \) |
| 29 | \( 1 - 3.38T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 - 5.26T + 37T^{2} \) |
| 41 | \( 1 + 6.09T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 0.403T + 47T^{2} \) |
| 53 | \( 1 + 5.88T + 53T^{2} \) |
| 59 | \( 1 + 9.60T + 59T^{2} \) |
| 61 | \( 1 + 7.09T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 0.478T + 71T^{2} \) |
| 73 | \( 1 - 2.40T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 4.90T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−7.88452945363152081708365430271, −6.88524400381432345104979220303, −6.42894163391443658216681517295, −5.95363632985781821872970235126, −4.98441337762186510926826957948, −4.27106437084947844768804702335, −3.38109427701902601368972664823, −2.83240130784197051410394809298, −1.61564442318116889645919096568, −0.841527663918498789589108559110,
0.841527663918498789589108559110, 1.61564442318116889645919096568, 2.83240130784197051410394809298, 3.38109427701902601368972664823, 4.27106437084947844768804702335, 4.98441337762186510926826957948, 5.95363632985781821872970235126, 6.42894163391443658216681517295, 6.88524400381432345104979220303, 7.88452945363152081708365430271
