| L(s) = 1 | + 3-s + 5-s + 2.82·7-s + 9-s + 11-s − 0.828·13-s + 15-s − 0.828·17-s + 2.82·21-s + 5.65·23-s + 25-s + 27-s + 7.65·29-s − 5.65·31-s + 33-s + 2.82·35-s − 2·37-s − 0.828·39-s + 0.343·41-s − 4.48·43-s + 45-s − 5.65·47-s + 1.00·49-s − 0.828·51-s + 0.343·53-s + 55-s + 1.65·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.06·7-s + 0.333·9-s + 0.301·11-s − 0.229·13-s + 0.258·15-s − 0.200·17-s + 0.617·21-s + 1.17·23-s + 0.200·25-s + 0.192·27-s + 1.42·29-s − 1.01·31-s + 0.174·33-s + 0.478·35-s − 0.328·37-s − 0.132·39-s + 0.0535·41-s − 0.683·43-s + 0.149·45-s − 0.825·47-s + 0.142·49-s − 0.116·51-s + 0.0471·53-s + 0.134·55-s + 0.215·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.573949650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.573949650\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 2.82T + 7T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 0.343T + 53T^{2} \) |
| 59 | \( 1 - 1.65T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 - 1.65T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 - 8.82T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−9.522073637355782167193256485842, −8.772424451004353332428203715181, −8.156135354144020063853841902398, −7.23652406452635658149669934371, −6.46625789409511792453159245358, −5.23082063169615795843764705564, −4.63597315371272154765776586312, −3.43692563315925546103585352219, −2.32531556249722211068428673851, −1.31011591310812085888926468047,
1.31011591310812085888926468047, 2.32531556249722211068428673851, 3.43692563315925546103585352219, 4.63597315371272154765776586312, 5.23082063169615795843764705564, 6.46625789409511792453159245358, 7.23652406452635658149669934371, 8.156135354144020063853841902398, 8.772424451004353332428203715181, 9.522073637355782167193256485842
