| L(s) = 1 | + 2-s + 4-s + 0.852·5-s + 1.37·7-s + 8-s + 0.852·10-s + 0.216·11-s + 1.95·13-s + 1.37·14-s + 16-s + 7.71·17-s − 6.12·19-s + 0.852·20-s + 0.216·22-s − 2.41·23-s − 4.27·25-s + 1.95·26-s + 1.37·28-s + 3.94·29-s + 8.84·31-s + 32-s + 7.71·34-s + 1.17·35-s + 6.63·37-s − 6.12·38-s + 0.852·40-s − 3.73·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.381·5-s + 0.520·7-s + 0.353·8-s + 0.269·10-s + 0.0653·11-s + 0.542·13-s + 0.367·14-s + 0.250·16-s + 1.87·17-s − 1.40·19-s + 0.190·20-s + 0.0462·22-s − 0.503·23-s − 0.854·25-s + 0.383·26-s + 0.260·28-s + 0.732·29-s + 1.58·31-s + 0.176·32-s + 1.32·34-s + 0.198·35-s + 1.09·37-s − 0.992·38-s + 0.134·40-s − 0.583·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.150576316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.150576316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 0.852T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - 0.216T + 11T^{2} \) |
| 13 | \( 1 - 1.95T + 13T^{2} \) |
| 17 | \( 1 - 7.71T + 17T^{2} \) |
| 19 | \( 1 + 6.12T + 19T^{2} \) |
| 23 | \( 1 + 2.41T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 - 8.84T + 31T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 1.52T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 - 4.62T + 59T^{2} \) |
| 61 | \( 1 - 5.24T + 61T^{2} \) |
| 67 | \( 1 - 4.45T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 5.39T + 73T^{2} \) |
| 79 | \( 1 + 8.02T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−9.898090656062284920791505235101, −8.312397265980720854967727106809, −8.148103241784945991868102195601, −6.85742051322493317955671668727, −6.11457113008690823134019956155, −5.39648321276598064811947264955, −4.44989480317072977897687036539, −3.57659861593411134296650869381, −2.43939393371240377911331797436, −1.30189643017779921672537757922,
1.30189643017779921672537757922, 2.43939393371240377911331797436, 3.57659861593411134296650869381, 4.44989480317072977897687036539, 5.39648321276598064811947264955, 6.11457113008690823134019956155, 6.85742051322493317955671668727, 8.148103241784945991868102195601, 8.312397265980720854967727106809, 9.898090656062284920791505235101
