| L(s) = 1 | + 2-s − 2.16·3-s + 4-s + 4.16·5-s − 2.16·6-s − 0.683·7-s + 8-s + 1.68·9-s + 4.16·10-s − 11-s − 2.16·12-s + 2.68·13-s − 0.683·14-s − 9.01·15-s + 16-s + 3.36·17-s + 1.68·18-s − 19-s + 4.16·20-s + 1.48·21-s − 22-s − 1.36·23-s − 2.16·24-s + 12.3·25-s + 2.68·26-s + 2.84·27-s − 0.683·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.24·3-s + 0.5·4-s + 1.86·5-s − 0.883·6-s − 0.258·7-s + 0.353·8-s + 0.561·9-s + 1.31·10-s − 0.301·11-s − 0.624·12-s + 0.744·13-s − 0.182·14-s − 2.32·15-s + 0.250·16-s + 0.816·17-s + 0.396·18-s − 0.229·19-s + 0.931·20-s + 0.323·21-s − 0.213·22-s − 0.285·23-s − 0.441·24-s + 2.46·25-s + 0.526·26-s + 0.548·27-s − 0.129·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.872909514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.872909514\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.16T + 3T^{2} \) |
| 5 | \( 1 - 4.16T + 5T^{2} \) |
| 7 | \( 1 + 0.683T + 7T^{2} \) |
| 13 | \( 1 - 2.68T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 23 | \( 1 + 1.36T + 23T^{2} \) |
| 29 | \( 1 - 2.79T + 29T^{2} \) |
| 31 | \( 1 - 5.01T + 31T^{2} \) |
| 37 | \( 1 + 11.6T + 37T^{2} \) |
| 41 | \( 1 + 7.01T + 41T^{2} \) |
| 43 | \( 1 - 7.86T + 43T^{2} \) |
| 47 | \( 1 - 2.96T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 + 9.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 0.683T + 67T^{2} \) |
| 71 | \( 1 - 3.53T + 71T^{2} \) |
| 73 | \( 1 - 0.407T + 73T^{2} \) |
| 79 | \( 1 + 7.28T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.15318713195075482282247711275, −10.40866193169665718996320851585, −9.828055502274430804721902238363, −8.540511508341507227250707888885, −6.86228195712903422927850873640, −6.12474216890903363498684479580, −5.61935710735571675983611659647, −4.78676607391630482467669756030, −3.01107070329188969587230374623, −1.50771487045926067104229885805,
1.50771487045926067104229885805, 3.01107070329188969587230374623, 4.78676607391630482467669756030, 5.61935710735571675983611659647, 6.12474216890903363498684479580, 6.86228195712903422927850873640, 8.540511508341507227250707888885, 9.828055502274430804721902238363, 10.40866193169665718996320851585, 11.15318713195075482282247711275
