L(s) = 1 | + 0.912·3-s − 1.66·5-s − 2.16·9-s − 0.242·11-s − 5.76·13-s − 1.51·15-s + 5.76·17-s + 19-s + 6.61·23-s − 2.22·25-s − 4.71·27-s + 8.91·29-s − 5.93·31-s − 0.220·33-s − 9.80·37-s − 5.26·39-s − 2·41-s − 7.83·43-s + 3.60·45-s + 1.57·47-s + 5.26·51-s + 6.48·53-s + 0.403·55-s + 0.912·57-s + 1.07·59-s + 9.10·61-s + 9.60·65-s + ⋯ |
L(s) = 1 | + 0.526·3-s − 0.744·5-s − 0.722·9-s − 0.0730·11-s − 1.60·13-s − 0.392·15-s + 1.39·17-s + 0.229·19-s + 1.37·23-s − 0.445·25-s − 0.907·27-s + 1.65·29-s − 1.06·31-s − 0.0384·33-s − 1.61·37-s − 0.842·39-s − 0.312·41-s − 1.19·43-s + 0.538·45-s + 0.230·47-s + 0.736·51-s + 0.890·53-s + 0.0543·55-s + 0.120·57-s + 0.139·59-s + 1.16·61-s + 1.19·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.444294550\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.444294550\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 0.912T + 3T^{2} \) |
| 5 | \( 1 + 1.66T + 5T^{2} \) |
| 11 | \( 1 + 0.242T + 11T^{2} \) |
| 13 | \( 1 + 5.76T + 13T^{2} \) |
| 17 | \( 1 - 5.76T + 17T^{2} \) |
| 23 | \( 1 - 6.61T + 23T^{2} \) |
| 29 | \( 1 - 8.91T + 29T^{2} \) |
| 31 | \( 1 + 5.93T + 31T^{2} \) |
| 37 | \( 1 + 9.80T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 7.83T + 43T^{2} \) |
| 47 | \( 1 - 1.57T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 - 1.07T + 59T^{2} \) |
| 61 | \( 1 - 9.10T + 61T^{2} \) |
| 67 | \( 1 - 8.02T + 67T^{2} \) |
| 71 | \( 1 - 1.22T + 71T^{2} \) |
| 73 | \( 1 - 0.801T + 73T^{2} \) |
| 79 | \( 1 + 7.58T + 79T^{2} \) |
| 83 | \( 1 + 8.91T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 1.11T + 97T^{2} \) |
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
−7.980742566578950828120203936553, −7.19894385785835669691935238223, −6.85853974334204954132604138828, −5.44592272442077110596749590912, −5.27158092336027634226410497901, −4.25827484856800844191050663282, −3.29250820079081998916134045192, −2.95743450417319163042219427380, −1.91874588274641452236736494683, −0.56757633580984542772623305889,
0.56757633580984542772623305889, 1.91874588274641452236736494683, 2.95743450417319163042219427380, 3.29250820079081998916134045192, 4.25827484856800844191050663282, 5.27158092336027634226410497901, 5.44592272442077110596749590912, 6.85853974334204954132604138828, 7.19894385785835669691935238223, 7.980742566578950828120203936553