| L(s) = 1 | + 2.65·2-s − 10.3·3-s − 0.959·4-s − 27.3·6-s − 23.7·8-s + 79.2·9-s + 9.89·12-s + 86.8·13-s − 55.3·16-s + 210.·18-s + 245.·24-s − 125·25-s + 230.·26-s − 538.·27-s − 24.7·29-s + 147.·31-s + 43.1·32-s − 76.0·36-s − 895.·39-s + 52.8·41-s − 580.·47-s + 570.·48-s − 343·49-s − 331.·50-s − 83.3·52-s − 1.42e3·54-s − 65.5·58-s + ⋯ |
| L(s) = 1 | + 0.938·2-s − 1.98·3-s − 0.119·4-s − 1.86·6-s − 1.05·8-s + 2.93·9-s + 0.238·12-s + 1.85·13-s − 0.865·16-s + 2.75·18-s + 2.08·24-s − 25-s + 1.73·26-s − 3.83·27-s − 0.158·29-s + 0.852·31-s + 0.238·32-s − 0.352·36-s − 3.67·39-s + 0.201·41-s − 1.80·47-s + 1.71·48-s − 49-s − 0.938·50-s − 0.222·52-s − 3.59·54-s − 0.148·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) |
| good | 2 | \( 1 - 2.65T + 8T^{2} \) |
| 3 | \( 1 + 10.3T + 27T^{2} \) |
| 5 | \( 1 + 125T^{2} \) |
| 7 | \( 1 + 343T^{2} \) |
| 11 | \( 1 + 1.33e3T^{2} \) |
| 13 | \( 1 - 86.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 29 | \( 1 + 24.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 5.06e4T^{2} \) |
| 41 | \( 1 - 52.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 + 580.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 + 396T + 2.05e5T^{2} \) |
| 61 | \( 1 + 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 779.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 812.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.12e5T^{2} \) |
| 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
−10.30792866267953517781133355327, −9.375203730262149961250469711864, −8.036041524584499579463020397091, −6.55672216130415090302545912052, −6.13306054101133600683790892643, −5.33942801291885513262722420341, −4.45490125756870609383157900573, −3.62451362876479624388955763697, −1.33146011199116449577164528096, 0,
1.33146011199116449577164528096, 3.62451362876479624388955763697, 4.45490125756870609383157900573, 5.33942801291885513262722420341, 6.13306054101133600683790892643, 6.55672216130415090302545912052, 8.036041524584499579463020397091, 9.375203730262149961250469711864, 10.30792866267953517781133355327
