| L(s) = 1 | + 2-s + 3-s + 4-s + 3.61·5-s + 6-s − 0.618·7-s + 8-s + 9-s + 3.61·10-s + 12-s − 3.23·13-s − 0.618·14-s + 3.61·15-s + 16-s − 4·17-s + 18-s + 5.70·19-s + 3.61·20-s − 0.618·21-s − 5.70·23-s + 24-s + 8.09·25-s − 3.23·26-s + 27-s − 0.618·28-s − 6.85·29-s + 3.61·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.61·5-s + 0.408·6-s − 0.233·7-s + 0.353·8-s + 0.333·9-s + 1.14·10-s + 0.288·12-s − 0.897·13-s − 0.165·14-s + 0.934·15-s + 0.250·16-s − 0.970·17-s + 0.235·18-s + 1.30·19-s + 0.809·20-s − 0.134·21-s − 1.19·23-s + 0.204·24-s + 1.61·25-s − 0.634·26-s + 0.192·27-s − 0.116·28-s − 1.27·29-s + 0.660·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.376188057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.376188057\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3.61T + 5T^{2} \) |
| 7 | \( 1 + 0.618T + 7T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 + 6.85T + 29T^{2} \) |
| 31 | \( 1 + 3.38T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 - 4.76T + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 - 1.38T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 6.85T + 79T^{2} \) |
| 83 | \( 1 + 1.61T + 83T^{2} \) |
| 89 | \( 1 - 6.76T + 89T^{2} \) |
| 97 | \( 1 + 9.61T + 97T^{2} \) |
| 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
−10.19390295091776595577392234564, −9.624203074422756411359389771397, −8.931569131370817837004739337838, −7.58419641060174188013045369730, −6.78966039490445011091561400575, −5.78254389319409234399629203010, −5.12874678394317856664033748073, −3.83712078997529041794991916724, −2.58707467631099493300437257464, −1.84701176515998251819044045617,
1.84701176515998251819044045617, 2.58707467631099493300437257464, 3.83712078997529041794991916724, 5.12874678394317856664033748073, 5.78254389319409234399629203010, 6.78966039490445011091561400575, 7.58419641060174188013045369730, 8.931569131370817837004739337838, 9.624203074422756411359389771397, 10.19390295091776595577392234564
