L(s) = 1 | + (4.17 + 3.09i)3-s + (−4.27 + 7.41i)5-s + (6.41 − 17.3i)7-s + (7.89 + 25.8i)9-s + (53.8 − 31.0i)11-s − 61.7i·13-s + (−40.7 + 17.7i)15-s + (13.3 + 23.0i)17-s + (58.6 + 33.8i)19-s + (80.4 − 52.7i)21-s + (45.8 + 26.4i)23-s + (25.8 + 44.8i)25-s + (−46.8 + 132. i)27-s + 55.1i·29-s + (134. − 77.4i)31-s + ⋯ |
L(s) = 1 | + (0.803 + 0.594i)3-s + (−0.382 + 0.662i)5-s + (0.346 − 0.938i)7-s + (0.292 + 0.956i)9-s + (1.47 − 0.852i)11-s − 1.31i·13-s + (−0.701 + 0.305i)15-s + (0.190 + 0.329i)17-s + (0.707 + 0.408i)19-s + (0.836 − 0.547i)21-s + (0.415 + 0.239i)23-s + (0.207 + 0.358i)25-s + (−0.333 + 0.942i)27-s + 0.352i·29-s + (0.776 − 0.448i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.689570061\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.689570061\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.17 - 3.09i)T \) |
| 7 | \( 1 + (-6.41 + 17.3i)T \) |
good | 5 | \( 1 + (4.27 - 7.41i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-53.8 + 31.0i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 61.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-13.3 - 23.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-58.6 - 33.8i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-45.8 - 26.4i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 55.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-134. + 77.4i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (157. - 273. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 351.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (115. - 200. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-232. + 134. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (9.14 + 15.8i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (72.3 + 41.7i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-64.7 - 112. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 804. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-370. + 213. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-609. + 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (386. - 670. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 848. iT - 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.96883474008263663042776952247, −10.34590332137531901799055148172, −9.379843371820207971427436242471, −8.264824378952081384610825108184, −7.60958346403226567833720371154, −6.49038163105796095409198294160, −5.01601328812020354567841940908, −3.66519198870825963891806012489, −3.23468335193098158761302003609, −1.19317774680238368452660988515,
1.20908638358490707417876830360, 2.31498015200345665242835092360, 3.88514211928309372891457792124, 4.88808186402206701885953431704, 6.47577632194577745475356215942, 7.22096157795062236002560827709, 8.452748973319153613512028799264, 9.052901802132866146245525595471, 9.653848107118094730481671259973, 11.55448370244565840484438467787