| L(s) = 1 | − 3·3-s + 5·5-s + 7·7-s + 9·9-s − 22·11-s − 44·13-s − 15·15-s − 110·17-s + 22·19-s − 21·21-s + 36·23-s + 25·25-s − 27·27-s − 122·29-s + 186·31-s + 66·33-s + 35·35-s + 306·37-s + 132·39-s − 330·41-s − 20·43-s + 45·45-s + 64·47-s + 49·49-s + 330·51-s + 504·53-s − 110·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.938·13-s − 0.258·15-s − 1.56·17-s + 0.265·19-s − 0.218·21-s + 0.326·23-s + 1/5·25-s − 0.192·27-s − 0.781·29-s + 1.07·31-s + 0.348·33-s + 0.169·35-s + 1.35·37-s + 0.541·39-s − 1.25·41-s − 0.0709·43-s + 0.149·45-s + 0.198·47-s + 1/7·49-s + 0.906·51-s + 1.30·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.435948214\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.435948214\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 44 T + p^{3} T^{2} \) |
| 17 | \( 1 + 110 T + p^{3} T^{2} \) |
| 19 | \( 1 - 22 T + p^{3} T^{2} \) |
| 23 | \( 1 - 36 T + p^{3} T^{2} \) |
| 29 | \( 1 + 122 T + p^{3} T^{2} \) |
| 31 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 37 | \( 1 - 306 T + p^{3} T^{2} \) |
| 41 | \( 1 + 330 T + p^{3} T^{2} \) |
| 43 | \( 1 + 20 T + p^{3} T^{2} \) |
| 47 | \( 1 - 64 T + p^{3} T^{2} \) |
| 53 | \( 1 - 504 T + p^{3} T^{2} \) |
| 59 | \( 1 - 560 T + p^{3} T^{2} \) |
| 61 | \( 1 + 418 T + p^{3} T^{2} \) |
| 67 | \( 1 - 452 T + p^{3} T^{2} \) |
| 71 | \( 1 - 146 T + p^{3} T^{2} \) |
| 73 | \( 1 + 236 T + p^{3} T^{2} \) |
| 79 | \( 1 + 536 T + p^{3} T^{2} \) |
| 83 | \( 1 - 92 T + p^{3} T^{2} \) |
| 89 | \( 1 + 574 T + p^{3} T^{2} \) |
| 97 | \( 1 - 184 T + p^{3} T^{2} \) |
| 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
−9.049951824571727620459305467748, −8.189782623832962638770063802368, −7.26713209556370263704708585391, −6.61427207096965207835012797895, −5.65046170029937008935989659359, −4.94038684507315154355779077149, −4.23161088101228542227330223250, −2.75364579503596591282805713007, −1.93984768967473906031030659086, −0.57379796557240756967202464938,
0.57379796557240756967202464938, 1.93984768967473906031030659086, 2.75364579503596591282805713007, 4.23161088101228542227330223250, 4.94038684507315154355779077149, 5.65046170029937008935989659359, 6.61427207096965207835012797895, 7.26713209556370263704708585391, 8.189782623832962638770063802368, 9.049951824571727620459305467748
