L(s) = 1 | − 2-s − 3-s + 4-s − 0.186·5-s + 6-s − 2.28·7-s − 8-s + 9-s + 0.186·10-s − 2.54·11-s − 12-s − 3.08·13-s + 2.28·14-s + 0.186·15-s + 16-s + 0.350·17-s − 18-s − 19-s − 0.186·20-s + 2.28·21-s + 2.54·22-s + 3.42·23-s + 24-s − 4.96·25-s + 3.08·26-s − 27-s − 2.28·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0835·5-s + 0.408·6-s − 0.862·7-s − 0.353·8-s + 0.333·9-s + 0.0590·10-s − 0.767·11-s − 0.288·12-s − 0.854·13-s + 0.609·14-s + 0.0482·15-s + 0.250·16-s + 0.0849·17-s − 0.235·18-s − 0.229·19-s − 0.0417·20-s + 0.497·21-s + 0.542·22-s + 0.714·23-s + 0.204·24-s − 0.993·25-s + 0.604·26-s − 0.192·27-s − 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3584181443\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3584181443\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 + 0.186T + 5T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 13 | \( 1 + 3.08T + 13T^{2} \) |
| 17 | \( 1 - 0.350T + 17T^{2} \) |
| 23 | \( 1 - 3.42T + 23T^{2} \) |
| 29 | \( 1 + 3.84T + 29T^{2} \) |
| 31 | \( 1 - 0.257T + 31T^{2} \) |
| 37 | \( 1 + 0.640T + 37T^{2} \) |
| 41 | \( 1 + 5.74T + 41T^{2} \) |
| 43 | \( 1 - 6.19T + 43T^{2} \) |
| 47 | \( 1 + 6.20T + 47T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 3.02T + 61T^{2} \) |
| 67 | \( 1 + 0.411T + 67T^{2} \) |
| 71 | \( 1 - 4.85T + 71T^{2} \) |
| 73 | \( 1 + 7.21T + 73T^{2} \) |
| 79 | \( 1 + 1.68T + 79T^{2} \) |
| 83 | \( 1 + 7.08T + 83T^{2} \) |
| 89 | \( 1 - 1.55T + 89T^{2} \) |
| 97 | \( 1 + 5.00T + 97T^{2} \) |
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
−7.944602682182720359582135842202, −7.40337828078126631752113921689, −6.76140190964852068746015329971, −6.03396480970899319339140973023, −5.35949348532548447812477627259, −4.55112501102123652759910108908, −3.47984632966418002176211248439, −2.69601434949686808590367355545, −1.72312463618480209611960700353, −0.34699169338110983737723536800,
0.34699169338110983737723536800, 1.72312463618480209611960700353, 2.69601434949686808590367355545, 3.47984632966418002176211248439, 4.55112501102123652759910108908, 5.35949348532548447812477627259, 6.03396480970899319339140973023, 6.76140190964852068746015329971, 7.40337828078126631752113921689, 7.944602682182720359582135842202