| L(s) = 1 | + (−4.5 + 7.79i)3-s + (45.8 + 79.4i)5-s + (−108. + 70.4i)7-s + (−40.5 − 70.1i)9-s + (−271. + 470. i)11-s − 28.2·13-s − 825.·15-s + (−509. + 883. i)17-s + (91.8 + 159. i)19-s + (−59.3 − 1.16e3i)21-s + (−2.31e3 − 4.00e3i)23-s + (−2.64e3 + 4.57e3i)25-s + 729·27-s + 3.33e3·29-s + (−4.13e3 + 7.16e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.288 + 0.499i)3-s + (0.820 + 1.42i)5-s + (−0.839 + 0.543i)7-s + (−0.166 − 0.288i)9-s + (−0.676 + 1.17i)11-s − 0.0463·13-s − 0.947·15-s + (−0.427 + 0.741i)17-s + (0.0583 + 0.101i)19-s + (−0.0293 − 0.576i)21-s + (−0.910 − 1.57i)23-s + (−0.845 + 1.46i)25-s + 0.192·27-s + 0.736·29-s + (−0.773 + 1.33i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7995178678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7995178678\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (108. - 70.4i)T \) |
| good | 5 | \( 1 + (-45.8 - 79.4i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (271. - 470. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 28.2T + 3.71e5T^{2} \) |
| 17 | \( 1 + (509. - 883. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-91.8 - 159. i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (2.31e3 + 4.00e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 - 3.33e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (4.13e3 - 7.16e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (958. + 1.66e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.88e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.28e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.26e4 - 2.19e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-2.94e3 + 5.09e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.83e4 + 3.17e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.82e4 - 3.16e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.39e4 + 2.42e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 3.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.56e4 - 2.71e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.86e4 + 6.70e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 7.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (7.17e4 + 1.24e5i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 5.06e4T + 8.58e9T^{2} \) |
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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
−10.98607573734951280519680551978, −10.28342480126655402820059639254, −9.871131031343223951492683145591, −8.748105432933185677547882408528, −7.25415619927776115670868954851, −6.41870818491672751214113492692, −5.71566684334943656197659125632, −4.31977901452482167214618125124, −2.91906243856766077526836719754, −2.16634274779953633967989592300,
0.23134949975945789309820884796, 1.05546475114740820952051868262, 2.49544893575436718211094658698, 4.05127274275756478480426995826, 5.46353050204842946574323718374, 5.88811354772291236129608033662, 7.22054793908849303123927300984, 8.264161729877030046730107066746, 9.237264075694891951912050550877, 9.932536395285699172625722647390
