L(s) = 1 | + 2-s − 0.618·3-s + 4-s + 5-s − 0.618·6-s − 1.38·7-s + 8-s − 2.61·9-s + 10-s − 11-s − 0.618·12-s − 1.38·14-s − 0.618·15-s + 16-s − 1.61·17-s − 2.61·18-s − 1.14·19-s + 20-s + 0.854·21-s − 22-s + 4·23-s − 0.618·24-s + 25-s + 3.47·27-s − 1.38·28-s + 5.70·29-s − 0.618·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.356·3-s + 0.5·4-s + 0.447·5-s − 0.252·6-s − 0.522·7-s + 0.353·8-s − 0.872·9-s + 0.316·10-s − 0.301·11-s − 0.178·12-s − 0.369·14-s − 0.159·15-s + 0.250·16-s − 0.392·17-s − 0.617·18-s − 0.262·19-s + 0.223·20-s + 0.186·21-s − 0.213·22-s + 0.834·23-s − 0.126·24-s + 0.200·25-s + 0.668·27-s − 0.261·28-s + 1.05·29-s − 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 7 | \( 1 + 1.38T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 5.70T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 47 | \( 1 - 2.61T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 3.38T + 59T^{2} \) |
| 61 | \( 1 + 0.763T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 4.76T + 73T^{2} \) |
| 79 | \( 1 + 3.85T + 79T^{2} \) |
| 83 | \( 1 + 0.326T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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
−7.893762219692713699672151884124, −6.83232665558455743027998532097, −6.44641819192039290429682302087, −5.68493143600603102001426840452, −5.07116780136841738459498628075, −4.32948350106586898033789535539, −3.13752735890126713284138139629, −2.74299983676129731826903276534, −1.51759811197994032048022505944, 0,
1.51759811197994032048022505944, 2.74299983676129731826903276534, 3.13752735890126713284138139629, 4.32948350106586898033789535539, 5.07116780136841738459498628075, 5.68493143600603102001426840452, 6.44641819192039290429682302087, 6.83232665558455743027998532097, 7.893762219692713699672151884124