| L(s) = 1 | + 5.13·2-s + 3·3-s + 18.3·4-s − 11.5·5-s + 15.3·6-s + 53.1·8-s + 9·9-s − 59.1·10-s − 11·11-s + 55.0·12-s − 63.3·13-s − 34.5·15-s + 125.·16-s − 76.6·17-s + 46.1·18-s − 125.·19-s − 211.·20-s − 56.4·22-s − 59.1·23-s + 159.·24-s + 7.93·25-s − 324.·26-s + 27·27-s − 93.6·29-s − 177.·30-s + 317.·31-s + 220.·32-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 0.577·3-s + 2.29·4-s − 1.03·5-s + 1.04·6-s + 2.34·8-s + 0.333·9-s − 1.87·10-s − 0.301·11-s + 1.32·12-s − 1.35·13-s − 0.595·15-s + 1.96·16-s − 1.09·17-s + 0.604·18-s − 1.51·19-s − 2.36·20-s − 0.547·22-s − 0.535·23-s + 1.35·24-s + 0.0634·25-s − 2.45·26-s + 0.192·27-s − 0.599·29-s − 1.08·30-s + 1.84·31-s + 1.22·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 5.13T + 8T^{2} \) |
| 5 | \( 1 + 11.5T + 125T^{2} \) |
| 13 | \( 1 + 63.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 76.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 59.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 93.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 317.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.39T + 5.06e4T^{2} \) |
| 41 | \( 1 - 131.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 247.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 509.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 417.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 353.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 98.3T + 2.26e5T^{2} \) |
| 67 | \( 1 - 933.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 16.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 722.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 541.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.35e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 140.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 342.T + 9.12e5T^{2} \) |
| 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
−8.241359905180668614113164853406, −7.70778374687273731514401909449, −6.81445080630678539223092507168, −6.19207161809064067280986687326, −4.87295652134117353414616104861, −4.44523998198888601211750961396, −3.73210068629293845001901594138, −2.69069810019385223975907348424, −2.08578209500129275481962360134, 0,
2.08578209500129275481962360134, 2.69069810019385223975907348424, 3.73210068629293845001901594138, 4.44523998198888601211750961396, 4.87295652134117353414616104861, 6.19207161809064067280986687326, 6.81445080630678539223092507168, 7.70778374687273731514401909449, 8.241359905180668614113164853406
