L(s) = 1 | + (−2.34 + 4.05i)5-s + (−10.9 + 14.9i)7-s + (−16.1 + 9.30i)11-s + (44.1 + 25.4i)13-s + 112.·17-s − 111. i·19-s + (124. + 71.8i)23-s + (51.5 + 89.2i)25-s + (−206. + 119. i)29-s + (−179. − 103. i)31-s + (−35.0 − 79.3i)35-s − 227.·37-s + (−133. + 230. i)41-s + (−170. − 294. i)43-s + (111. + 193. i)47-s + ⋯ |
L(s) = 1 | + (−0.209 + 0.362i)5-s + (−0.590 + 0.807i)7-s + (−0.441 + 0.255i)11-s + (0.941 + 0.543i)13-s + 1.60·17-s − 1.34i·19-s + (1.12 + 0.651i)23-s + (0.412 + 0.713i)25-s + (−1.32 + 0.764i)29-s + (−1.04 − 0.602i)31-s + (−0.169 − 0.383i)35-s − 1.01·37-s + (−0.507 + 0.878i)41-s + (−0.603 − 1.04i)43-s + (0.347 + 0.601i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.879 - 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.022521933\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.022521933\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (10.9 - 14.9i)T \) |
good | 5 | \( 1 + (2.34 - 4.05i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (16.1 - 9.30i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-44.1 - 25.4i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 - 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 111. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-124. - 71.8i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (206. - 119. i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (179. + 103. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (133. - 230. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (170. + 294. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-111. - 193. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 - 547. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-43.9 + 76.1i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (312. - 180. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (372. - 645. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 135. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 467. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + (-192. - 332. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (597. + 1.03e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.07e3 + 617. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
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
−10.35163108240110499772020595485, −9.211458849879845610020063249043, −8.915587917643253020823728036166, −7.53232838102501502766988028318, −6.96378365713962722761990571418, −5.78965219314120976810710131702, −5.12572537091719765841813255468, −3.59544077258230159937089328353, −2.92204329579666092614356788459, −1.45775712720397272661417623339,
0.29316645728850382299030183683, 1.41465024446327916807557251373, 3.24675155950768255122320194128, 3.79100875239032924409617621969, 5.18025761084864294336449647973, 5.96072515438575404204711201717, 7.05262955755907950787540715161, 7.931183957514355024705774606119, 8.594524735095409899949646727825, 9.748202598506947551208156075705