| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 2.37·7-s + 8-s + 9-s + 10-s + 6.37·11-s + 12-s − 2·13-s − 2.37·14-s + 15-s + 16-s + 6.74·17-s + 18-s − 6.37·19-s + 20-s − 2.37·21-s + 6.37·22-s − 2.37·23-s + 24-s + 25-s − 2·26-s + 27-s − 2.37·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.896·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.92·11-s + 0.288·12-s − 0.554·13-s − 0.634·14-s + 0.258·15-s + 0.250·16-s + 1.63·17-s + 0.235·18-s − 1.46·19-s + 0.223·20-s − 0.517·21-s + 1.35·22-s − 0.494·23-s + 0.204·24-s + 0.200·25-s − 0.392·26-s + 0.192·27-s − 0.448·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.178869393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.178869393\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 6.74T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 + 2.37T + 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 6.37T + 43T^{2} \) |
| 47 | \( 1 - 4.74T + 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 0.744T + 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 - 9.11T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
| 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
−9.845877171217146095031777213704, −9.456302010721688682340875744034, −8.432811589767721535782102862692, −7.35335038309971222476904616927, −6.43135534110010203914528061429, −5.96903423339873740311733271192, −4.52857870310880117502749956825, −3.71974408397570493107409839858, −2.79987541782877139287620011463, −1.49686357749362007261221879632,
1.49686357749362007261221879632, 2.79987541782877139287620011463, 3.71974408397570493107409839858, 4.52857870310880117502749956825, 5.96903423339873740311733271192, 6.43135534110010203914528061429, 7.35335038309971222476904616927, 8.432811589767721535782102862692, 9.456302010721688682340875744034, 9.845877171217146095031777213704
