| L(s) = 1 | + 2.36·2-s − 2.42·4-s + 6.42·5-s + 29.4·7-s − 24.6·8-s + 15.1·10-s − 0.624·11-s + 69.6·14-s − 38.7·16-s − 87.7·17-s − 82.8·19-s − 15.5·20-s − 1.47·22-s + 74.7·23-s − 83.7·25-s − 71.4·28-s − 226.·29-s − 173.·31-s + 105.·32-s − 207.·34-s + 189.·35-s − 112.·37-s − 195.·38-s − 158.·40-s − 267.·41-s + 383.·43-s + 1.51·44-s + ⋯ |
| L(s) = 1 | + 0.835·2-s − 0.302·4-s + 0.574·5-s + 1.59·7-s − 1.08·8-s + 0.479·10-s − 0.0171·11-s + 1.32·14-s − 0.605·16-s − 1.25·17-s − 0.999·19-s − 0.173·20-s − 0.0142·22-s + 0.678·23-s − 0.670·25-s − 0.482·28-s − 1.44·29-s − 1.00·31-s + 0.582·32-s − 1.04·34-s + 0.914·35-s − 0.497·37-s − 0.834·38-s − 0.624·40-s − 1.01·41-s + 1.35·43-s + 0.00518·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.36T + 8T^{2} \) |
| 5 | \( 1 - 6.42T + 125T^{2} \) |
| 7 | \( 1 - 29.4T + 343T^{2} \) |
| 11 | \( 1 + 0.624T + 1.33e3T^{2} \) |
| 17 | \( 1 + 87.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 82.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 74.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 112.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 267.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 337.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 146.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 529.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 203.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 121.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 661.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 167.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 101.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 506.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.40e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.90e3T + 9.12e5T^{2} \) |
| 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
−8.900183533793968209737790760981, −7.932677349973326798111616589450, −6.98024792562706225848768568502, −5.89486899117122821424465377837, −5.31553799592030991497642614208, −4.50261261544489429955864506275, −3.88360343977545662423930801983, −2.43849276021562390482553847028, −1.63833666428428244574655401783, 0,
1.63833666428428244574655401783, 2.43849276021562390482553847028, 3.88360343977545662423930801983, 4.50261261544489429955864506275, 5.31553799592030991497642614208, 5.89486899117122821424465377837, 6.98024792562706225848768568502, 7.932677349973326798111616589450, 8.900183533793968209737790760981
