| L(s) = 1 | − 4.51·3-s + 2.23i·5-s − 1.75i·7-s + 11.4·9-s − 18.1i·11-s + 15.0·13-s − 10.0i·15-s − 14.5i·17-s + 3.31i·19-s + 7.92i·21-s + (−19.3 − 12.4i)23-s − 5.00·25-s − 10.8·27-s − 10.7·29-s + 52.4·31-s + ⋯ |
| L(s) = 1 | − 1.50·3-s + 0.447i·5-s − 0.250i·7-s + 1.26·9-s − 1.64i·11-s + 1.16·13-s − 0.673i·15-s − 0.854i·17-s + 0.174i·19-s + 0.377i·21-s + (−0.840 − 0.541i)23-s − 0.200·25-s − 0.401·27-s − 0.371·29-s + 1.69·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.541 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7477549997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7477549997\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (19.3 + 12.4i)T \) |
| good | 3 | \( 1 + 4.51T + 9T^{2} \) |
| 7 | \( 1 + 1.75iT - 49T^{2} \) |
| 11 | \( 1 + 18.1iT - 121T^{2} \) |
| 13 | \( 1 - 15.0T + 169T^{2} \) |
| 17 | \( 1 + 14.5iT - 289T^{2} \) |
| 19 | \( 1 - 3.31iT - 361T^{2} \) |
| 29 | \( 1 + 10.7T + 841T^{2} \) |
| 31 | \( 1 - 52.4T + 961T^{2} \) |
| 37 | \( 1 - 40.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 0.449T + 1.68e3T^{2} \) |
| 43 | \( 1 + 10.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 72.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 11.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 39.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 87.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 59.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 54.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 94.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 30.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 98.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 133. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 60.8iT - 9.40e3T^{2} \) |
| 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
−8.666655860920271073031461124720, −8.054499842993987076489454514959, −6.86925708787185333343587697305, −6.24232102090892647948477850863, −5.81125635256861086022032426955, −4.86630366688583496713404083068, −3.85587549949096078472644860993, −2.89018735586051735660160941229, −1.17226204091957520727690770645, −0.30950985289801993967858390062,
1.09029560883275765415169967668, 2.06703566996453428743588886746, 3.82694658396161636850062194861, 4.57831568364108805266684749283, 5.34579506258495026257319393945, 6.12058672673938712094622659237, 6.67940229915195524653095326956, 7.69175873352092583349629604973, 8.514885499131676059253247401376, 9.552830236014179696784362658762
