| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s + 11-s + 13-s + 14-s + 16-s − 7·19-s + 20-s − 22-s − 3·23-s + 25-s − 26-s − 28-s − 8·29-s + 5·31-s − 32-s − 35-s − 5·37-s + 7·38-s − 40-s − 10·41-s + 2·43-s + 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 1.60·19-s + 0.223·20-s − 0.213·22-s − 0.625·23-s + 1/5·25-s − 0.196·26-s − 0.188·28-s − 1.48·29-s + 0.898·31-s − 0.176·32-s − 0.169·35-s − 0.821·37-s + 1.13·38-s − 0.158·40-s − 1.56·41-s + 0.304·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
| 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
−13.04214766590461, −12.43886716990282, −12.05738365554378, −11.38792495499901, −11.17801333401427, −10.46484757370320, −10.20919319758852, −9.797456452536018, −9.211660779749745, −8.838113730130296, −8.439517140026765, −7.892935247365018, −7.463143623698239, −6.706730014897277, −6.401493574137261, −6.185875761986189, −5.396006662880862, −4.957590288499851, −4.241179176196488, −3.659862073752530, −3.264000422971378, −2.453987822476451, −1.913522419018372, −1.605806425544222, −0.6418492529897764, 0,
0.6418492529897764, 1.605806425544222, 1.913522419018372, 2.453987822476451, 3.264000422971378, 3.659862073752530, 4.241179176196488, 4.957590288499851, 5.396006662880862, 6.185875761986189, 6.401493574137261, 6.706730014897277, 7.463143623698239, 7.892935247365018, 8.439517140026765, 8.838113730130296, 9.211660779749745, 9.797456452536018, 10.20919319758852, 10.46484757370320, 11.17801333401427, 11.38792495499901, 12.05738365554378, 12.43886716990282, 13.04214766590461
