| L(s) = 1 | + 20i·7-s − 16·11-s − 58i·13-s − 38i·17-s − 4·19-s − 80i·23-s + 82·29-s − 8·31-s + 426i·37-s + 246·41-s + 524i·43-s + 464i·47-s − 57·49-s − 702i·53-s − 592·59-s + ⋯ |
| L(s) = 1 | + 1.07i·7-s − 0.438·11-s − 1.23i·13-s − 0.542i·17-s − 0.0482·19-s − 0.725i·23-s + 0.525·29-s − 0.0463·31-s + 1.89i·37-s + 0.937·41-s + 1.85i·43-s + 1.44i·47-s − 0.166·49-s − 1.81i·53-s − 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6243992827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6243992827\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 20iT - 343T^{2} \) |
| 11 | \( 1 + 16T + 1.33e3T^{2} \) |
| 13 | \( 1 + 58iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 38iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 80iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 82T + 2.43e4T^{2} \) |
| 31 | \( 1 + 8T + 2.97e4T^{2} \) |
| 37 | \( 1 - 426iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 246T + 6.89e4T^{2} \) |
| 43 | \( 1 - 524iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 464iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 702iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 592T + 2.05e5T^{2} \) |
| 61 | \( 1 - 574T + 2.26e5T^{2} \) |
| 67 | \( 1 + 172iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 768T + 3.57e5T^{2} \) |
| 73 | \( 1 - 558iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 408T + 4.93e5T^{2} \) |
| 83 | \( 1 - 164iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 510T + 7.04e5T^{2} \) |
| 97 | \( 1 - 514iT - 9.12e5T^{2} \) |
| 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
−9.288114226828239293753977262184, −8.253321698021315909811104311238, −7.996808830802879189537972607747, −6.77886743514006506286687694441, −5.98064069019703983946956183829, −5.24497244656758590966007863730, −4.48329925214013270137423184135, −3.00211465037246109890376804432, −2.62864162346523547279153904977, −1.16557878145854026123175189443,
0.13793380440452283220858643500, 1.36665556468595375922304440596, 2.40845661680501109674956158571, 3.77859865402879561998910321737, 4.22040070059689329071241077562, 5.32678184117386842834354431399, 6.22347300470339390118252339297, 7.20899206776394810021968906343, 7.53046943296267950319079240310, 8.692750745535502815404172927081
