L(s) = 1 | + (−5.18 − 0.262i)3-s + (−2.24 − 3.88i)5-s + (9.71 − 15.7i)7-s + (26.8 + 2.72i)9-s + (20.2 + 11.7i)11-s − 5.91i·13-s + (10.6 + 20.7i)15-s + (−58.0 + 100. i)17-s + (−8.02 + 4.63i)19-s + (−54.5 + 79.2i)21-s + (107. − 62.1i)23-s + (52.4 − 90.7i)25-s + (−138. − 21.1i)27-s − 207. i·29-s + (−122. − 70.8i)31-s + ⋯ |
L(s) = 1 | + (−0.998 − 0.0504i)3-s + (−0.200 − 0.347i)5-s + (0.524 − 0.851i)7-s + (0.994 + 0.100i)9-s + (0.555 + 0.320i)11-s − 0.126i·13-s + (0.183 + 0.357i)15-s + (−0.828 + 1.43i)17-s + (−0.0969 + 0.0559i)19-s + (−0.566 + 0.823i)21-s + (0.976 − 0.563i)23-s + (0.419 − 0.726i)25-s + (−0.988 − 0.150i)27-s − 1.33i·29-s + (−0.711 − 0.410i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9477816048\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9477816048\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.18 + 0.262i)T \) |
| 7 | \( 1 + (-9.71 + 15.7i)T \) |
good | 5 | \( 1 + (2.24 + 3.88i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-20.2 - 11.7i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 5.91iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (58.0 - 100. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (8.02 - 4.63i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-107. + 62.1i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 207. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (122. + 70.8i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (149. + 259. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 508.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 391.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (40.2 + 69.7i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (258. + 149. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-102. + 177. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (543. - 313. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (51.3 - 89.0i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 46.9iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (228. + 131. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (533. + 924. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 270.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (443. + 768. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 219. iT - 9.12e5T^{2} \) |
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
−10.87162198420023083905194461488, −10.21192772726234136153546049673, −8.950944883025278726715186530328, −7.84447658534103596741246835273, −6.87152759686833439935841277654, −5.96746734830252364589365257844, −4.62731164286983349950134425775, −4.05021010508381613529282861377, −1.73730105119271177714333323216, −0.41909662787306163045479441614,
1.38321231661532605736560019866, 3.10155796143706961788567727477, 4.69450367763794350200374504211, 5.41587663853729881864585577911, 6.63170483998945979932457262151, 7.33004164485749619522414719342, 8.818318969179638769811371616825, 9.499092303714174984280120135422, 10.99134604778690551959171972942, 11.24133933816466703513078800779