| L(s) = 1 | + (−4.59 − 2.42i)3-s − 2.41i·5-s − 7i·7-s + (15.2 + 22.2i)9-s − 37.7·11-s − 24.8·13-s + (−5.83 + 11.0i)15-s + 22.2i·17-s − 68.4i·19-s + (−16.9 + 32.1i)21-s − 44.4·23-s + 119.·25-s + (−16.2 − 139. i)27-s + 178. i·29-s + 109. i·31-s + ⋯ |
| L(s) = 1 | + (−0.884 − 0.466i)3-s − 0.215i·5-s − 0.377i·7-s + (0.565 + 0.824i)9-s − 1.03·11-s − 0.529·13-s + (−0.100 + 0.190i)15-s + 0.317i·17-s − 0.826i·19-s + (−0.176 + 0.334i)21-s − 0.403·23-s + 0.953·25-s + (−0.116 − 0.993i)27-s + 1.14i·29-s + 0.636i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7401489284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7401489284\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.59 + 2.42i)T \) |
| 7 | \( 1 + 7iT \) |
| good | 5 | \( 1 + 2.41iT - 125T^{2} \) |
| 11 | \( 1 + 37.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 24.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 22.2iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 68.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 44.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 178. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 109. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 168.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 383. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 371. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 323.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 401. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 34.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 25.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 118. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 106.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.01e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 649. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 250.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 4.02iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 454.T + 9.12e5T^{2} \) |
| 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
−11.14007317035691097159761880780, −10.62170062377369790185128414058, −9.596417704151501572610900202641, −8.254225687752695130348031227648, −7.37612802735842116610081651945, −6.48792251886861889073098872686, −5.29366991859358663644506159146, −4.54895275449230744786170091566, −2.70449369670599026849114737275, −1.08602230662034672769811064608,
0.34547317786873830648959549179, 2.39893836580177432338743985545, 3.92093760213106499819544285067, 5.13368740011886426935399087314, 5.85963966416490293941682362380, 7.01942901159617954033934195857, 8.066699455208226930504750966213, 9.352612392590802727095601292171, 10.19120817805832744167353915363, 10.86538311243544291688056101861
