| L(s) = 1 | + 2-s + 2.43·3-s + 4-s − 5-s + 2.43·6-s − 3.35·7-s + 8-s + 2.93·9-s − 10-s − 3.03·11-s + 2.43·12-s + 13-s − 3.35·14-s − 2.43·15-s + 16-s − 4.12·17-s + 2.93·18-s − 2.24·19-s − 20-s − 8.18·21-s − 3.03·22-s − 2.50·23-s + 2.43·24-s + 25-s + 26-s − 0.158·27-s − 3.35·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.40·3-s + 0.5·4-s − 0.447·5-s + 0.994·6-s − 1.26·7-s + 0.353·8-s + 0.978·9-s − 0.316·10-s − 0.915·11-s + 0.703·12-s + 0.277·13-s − 0.897·14-s − 0.629·15-s + 0.250·16-s − 0.999·17-s + 0.691·18-s − 0.515·19-s − 0.223·20-s − 1.78·21-s − 0.647·22-s − 0.522·23-s + 0.497·24-s + 0.200·25-s + 0.196·26-s − 0.0304·27-s − 0.634·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 - 2.43T + 3T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 17 | \( 1 + 4.12T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 + 2.50T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 37 | \( 1 - 8.42T + 37T^{2} \) |
| 41 | \( 1 - 3.29T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 2.41T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 6.06T + 71T^{2} \) |
| 73 | \( 1 + 1.74T + 73T^{2} \) |
| 79 | \( 1 - 9.95T + 79T^{2} \) |
| 83 | \( 1 - 0.183T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 5.54T + 97T^{2} \) |
| 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
−7.914362136331158545876156548787, −7.53654996332636203658157455226, −6.56317914479734711214185186483, −5.98364943828795711269276144143, −4.86317830940873443169435975126, −3.92725483185964929202318235836, −3.45864112595461403444891569306, −2.67187855685231552205747493125, −2.00188938433334472994055470085, 0,
2.00188938433334472994055470085, 2.67187855685231552205747493125, 3.45864112595461403444891569306, 3.92725483185964929202318235836, 4.86317830940873443169435975126, 5.98364943828795711269276144143, 6.56317914479734711214185186483, 7.53654996332636203658157455226, 7.914362136331158545876156548787
