L(s) = 1 | + 2.63·2-s − 1.40·3-s + 4.91·4-s − 5-s − 3.68·6-s + 1.05·7-s + 7.67·8-s − 1.03·9-s − 2.63·10-s + 0.340·11-s − 6.88·12-s + 6.67·13-s + 2.77·14-s + 1.40·15-s + 10.3·16-s + 1.88·17-s − 2.73·18-s − 0.838·19-s − 4.91·20-s − 1.47·21-s + 0.894·22-s + 0.613·23-s − 10.7·24-s + 25-s + 17.5·26-s + 5.65·27-s + 5.18·28-s + ⋯ |
L(s) = 1 | + 1.85·2-s − 0.808·3-s + 2.45·4-s − 0.447·5-s − 1.50·6-s + 0.398·7-s + 2.71·8-s − 0.346·9-s − 0.831·10-s + 0.102·11-s − 1.98·12-s + 1.85·13-s + 0.741·14-s + 0.361·15-s + 2.58·16-s + 0.456·17-s − 0.643·18-s − 0.192·19-s − 1.09·20-s − 0.322·21-s + 0.190·22-s + 0.127·23-s − 2.19·24-s + 0.200·25-s + 3.44·26-s + 1.08·27-s + 0.979·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.061003265\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.061003265\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 3 | \( 1 + 1.40T + 3T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 - 0.340T + 11T^{2} \) |
| 13 | \( 1 - 6.67T + 13T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 + 0.838T + 19T^{2} \) |
| 23 | \( 1 - 0.613T + 23T^{2} \) |
| 29 | \( 1 - 1.63T + 29T^{2} \) |
| 31 | \( 1 - 6.39T + 31T^{2} \) |
| 37 | \( 1 + 2.44T + 37T^{2} \) |
| 41 | \( 1 + 3.67T + 41T^{2} \) |
| 43 | \( 1 + 4.17T + 43T^{2} \) |
| 47 | \( 1 - 0.850T + 47T^{2} \) |
| 53 | \( 1 + 7.75T + 53T^{2} \) |
| 59 | \( 1 - 2.57T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 + 7.56T + 71T^{2} \) |
| 73 | \( 1 + 6.53T + 73T^{2} \) |
| 79 | \( 1 + 0.258T + 79T^{2} \) |
| 83 | \( 1 + 5.90T + 83T^{2} \) |
| 89 | \( 1 - 3.73T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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
−10.33442095570422161346218943628, −8.659799146646023609808153400937, −7.892868603143158620817049629986, −6.64773160672109553411429813355, −6.23631973644495159662550491231, −5.39902657963243975543686351067, −4.68026195009056892785596141517, −3.75899605322407282769330893388, −2.96195387844818204586684967789, −1.38582518112777847860673063974,
1.38582518112777847860673063974, 2.96195387844818204586684967789, 3.75899605322407282769330893388, 4.68026195009056892785596141517, 5.39902657963243975543686351067, 6.23631973644495159662550491231, 6.64773160672109553411429813355, 7.892868603143158620817049629986, 8.659799146646023609808153400937, 10.33442095570422161346218943628