| L(s) = 1 | + 2-s + 1.27·3-s + 4-s + 1.53·5-s + 1.27·6-s + 1.94·7-s + 8-s − 1.38·9-s + 1.53·10-s + 0.441·11-s + 1.27·12-s − 2.11·13-s + 1.94·14-s + 1.94·15-s + 16-s − 3.82·17-s − 1.38·18-s + 5.94·19-s + 1.53·20-s + 2.47·21-s + 0.441·22-s + 5.71·23-s + 1.27·24-s − 2.65·25-s − 2.11·26-s − 5.57·27-s + 1.94·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.733·3-s + 0.5·4-s + 0.685·5-s + 0.518·6-s + 0.735·7-s + 0.353·8-s − 0.462·9-s + 0.484·10-s + 0.132·11-s + 0.366·12-s − 0.587·13-s + 0.520·14-s + 0.502·15-s + 0.250·16-s − 0.927·17-s − 0.326·18-s + 1.36·19-s + 0.342·20-s + 0.539·21-s + 0.0940·22-s + 1.19·23-s + 0.259·24-s − 0.530·25-s − 0.415·26-s − 1.07·27-s + 0.367·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.506697420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.506697420\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 1.27T + 3T^{2} \) |
| 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 - 1.94T + 7T^{2} \) |
| 11 | \( 1 - 0.441T + 11T^{2} \) |
| 13 | \( 1 + 2.11T + 13T^{2} \) |
| 17 | \( 1 + 3.82T + 17T^{2} \) |
| 19 | \( 1 - 5.94T + 19T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 - 2.73T + 29T^{2} \) |
| 31 | \( 1 - 9.87T + 31T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 3.84T + 43T^{2} \) |
| 47 | \( 1 - 7.68T + 47T^{2} \) |
| 53 | \( 1 - 1.72T + 53T^{2} \) |
| 59 | \( 1 - 2.05T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 1.67T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 9.88T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 3.55T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 0.593T + 97T^{2} \) |
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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
−8.918758878826557718197831463173, −7.942169711129800259959006210549, −7.40223574316020907292606531316, −6.41192109193991101255789538488, −5.63687291586803368862565294383, −4.89585516065346490036147933949, −4.13334787497297578444515970465, −2.86419344133991169010427846215, −2.49868633175519812193845113511, −1.27773793632809221690416142247,
1.27773793632809221690416142247, 2.49868633175519812193845113511, 2.86419344133991169010427846215, 4.13334787497297578444515970465, 4.89585516065346490036147933949, 5.63687291586803368862565294383, 6.41192109193991101255789538488, 7.40223574316020907292606531316, 7.942169711129800259959006210549, 8.918758878826557718197831463173
