L(s) = 1 | + (−4.5 + 7.79i)3-s + (−19.3 − 33.5i)5-s + (87.5 − 95.6i)7-s + (−40.5 − 70.1i)9-s + (−288. + 499. i)11-s + 391.·13-s + 348.·15-s + (664. − 1.15e3i)17-s + (471. + 816. i)19-s + (351. + 1.11e3i)21-s + (−816. − 1.41e3i)23-s + (812. − 1.40e3i)25-s + 729·27-s − 1.46e3·29-s + (−1.95e3 + 3.38e3i)31-s + ⋯ |
L(s) = 1 | + (−0.288 + 0.499i)3-s + (−0.346 − 0.599i)5-s + (0.674 − 0.737i)7-s + (−0.166 − 0.288i)9-s + (−0.718 + 1.24i)11-s + 0.642·13-s + 0.399·15-s + (0.557 − 0.966i)17-s + (0.299 + 0.518i)19-s + (0.174 + 0.550i)21-s + (−0.321 − 0.557i)23-s + (0.260 − 0.450i)25-s + 0.192·27-s − 0.323·29-s + (−0.365 + 0.633i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.087171410\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.087171410\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 - 7.79i)T \) |
| 7 | \( 1 + (-87.5 + 95.6i)T \) |
good | 5 | \( 1 + (19.3 + 33.5i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (288. - 499. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 - 391.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-664. + 1.15e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-471. - 816. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (816. + 1.41e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 1.46e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (1.95e3 - 3.38e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-8.15e3 - 1.41e4i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.31e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.47e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (3.40e3 + 5.90e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.00e3 + 1.74e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-2.57e4 + 4.45e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.05e4 + 3.55e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.52e4 + 4.38e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 3.99e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.78e4 + 4.82e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.15e4 + 5.46e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 4.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (7.84e3 + 1.35e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 3.12e3T + 8.58e9T^{2} \) |
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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
−10.32091345598006864867236017383, −9.784034311281721632784187758470, −8.441873915261240413220034636283, −7.73848275389500434204338445011, −6.62946723123400906210811468591, −5.04839945335906316653667957066, −4.68045492307606269617540877577, −3.40453930163319354171056230492, −1.63433161686096948174573854568, −0.31561707669727446388179123362,
1.22358426558276070032377914069, 2.60498203102772553350639142484, 3.74468213645877401036924822180, 5.44172579933310185463394540904, 5.95196967619012783855715792317, 7.28891728937252379464928818092, 8.124541244781304290015506780566, 8.866432142649006486993546397178, 10.34339461458288746168943816009, 11.27843118485047140817733345075