L(s) = 1 | − 0.0650·2-s − 3-s − 1.99·4-s + 2.65·5-s + 0.0650·6-s + 0.259·8-s + 9-s − 0.172·10-s − 5.47·11-s + 1.99·12-s + 3.40·13-s − 2.65·15-s + 3.97·16-s + 6.28·17-s − 0.0650·18-s + 4.25·19-s − 5.29·20-s + 0.355·22-s − 1.06·23-s − 0.259·24-s + 2.04·25-s − 0.221·26-s − 27-s − 6.10·29-s + 0.172·30-s + 10.4·31-s − 0.778·32-s + ⋯ |
L(s) = 1 | − 0.0459·2-s − 0.577·3-s − 0.997·4-s + 1.18·5-s + 0.0265·6-s + 0.0918·8-s + 0.333·9-s − 0.0545·10-s − 1.64·11-s + 0.576·12-s + 0.943·13-s − 0.685·15-s + 0.993·16-s + 1.52·17-s − 0.0153·18-s + 0.976·19-s − 1.18·20-s + 0.0758·22-s − 0.222·23-s − 0.0530·24-s + 0.408·25-s − 0.0434·26-s − 0.192·27-s − 1.13·29-s + 0.0315·30-s + 1.88·31-s − 0.137·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.515699794\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.515699794\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
good | 2 | \( 1 + 0.0650T + 2T^{2} \) |
| 5 | \( 1 - 2.65T + 5T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 - 3.40T + 13T^{2} \) |
| 17 | \( 1 - 6.28T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 + 1.06T + 23T^{2} \) |
| 29 | \( 1 + 6.10T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 - 3.83T + 37T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 8.14T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 6.51T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 4.82T + 73T^{2} \) |
| 79 | \( 1 - 9.32T + 79T^{2} \) |
| 83 | \( 1 - 5.03T + 83T^{2} \) |
| 89 | \( 1 + 8.62T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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
−7.940627445652079048651456077243, −7.65313700578150562165382192110, −6.34770705519093733768125529830, −5.76625025856994407798225759713, −5.30617977848672284578608859525, −4.76713530029188152344050149749, −3.61601062222478815607284944936, −2.85667607893493603101985021937, −1.61547584015165843537924531488, −0.71431518586463662859576292317,
0.71431518586463662859576292317, 1.61547584015165843537924531488, 2.85667607893493603101985021937, 3.61601062222478815607284944936, 4.76713530029188152344050149749, 5.30617977848672284578608859525, 5.76625025856994407798225759713, 6.34770705519093733768125529830, 7.65313700578150562165382192110, 7.940627445652079048651456077243