L(s) = 1 | + (47.4 − 82.2i)5-s + 189.·7-s + 530.·11-s + (−353. − 612. i)13-s + (764. − 1.32e3i)17-s + (654. − 1.43e3i)19-s + (−497. − 861. i)23-s + (−2.94e3 − 5.10e3i)25-s + (1.29e3 + 2.23e3i)29-s − 2.79e3·31-s + (9.01e3 − 1.56e4i)35-s + 7.23e3·37-s + (−2.18e3 + 3.77e3i)41-s + (−3.12e3 + 5.40e3i)43-s + (−1.23e4 − 2.13e4i)47-s + ⋯ |
L(s) = 1 | + (0.849 − 1.47i)5-s + 1.46·7-s + 1.32·11-s + (−0.580 − 1.00i)13-s + (0.641 − 1.11i)17-s + (0.415 − 0.909i)19-s + (−0.196 − 0.339i)23-s + (−0.943 − 1.63i)25-s + (0.284 + 0.493i)29-s − 0.521·31-s + (1.24 − 2.15i)35-s + 0.869·37-s + (−0.202 + 0.351i)41-s + (−0.257 + 0.445i)43-s + (−0.813 − 1.40i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.214 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(3.685468983\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.685468983\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-654. + 1.43e3i)T \) |
good | 5 | \( 1 + (-47.4 + 82.2i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 - 189.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 530.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (353. + 612. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-764. + 1.32e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 23 | \( 1 + (497. + 861. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-1.29e3 - 2.23e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 2.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.23e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (2.18e3 - 3.77e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (3.12e3 - 5.40e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.23e4 + 2.13e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.54e4 - 2.68e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.81e4 + 3.14e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.88e3 - 5.00e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-3.03e4 - 5.26e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (1.48e4 - 2.56e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (3.90e4 - 6.75e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.34e4 - 5.78e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 2.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-4.53e4 - 7.85e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + (3.07e4 - 5.32e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−9.389161501126694321541226248089, −8.662515021688905762137241257188, −7.937162115579723109011543806561, −6.86137536222830482460724880468, −5.40731728864796925264780071570, −5.14645454689593130229822592844, −4.20170018929066972394756808067, −2.53597002001404541343845478287, −1.31885320887556455323081311591, −0.815744886970347930006481232361,
1.54710663439217735863219270035, 1.97325616357226237304090279028, 3.41976001247719492298343638860, 4.38777001092229990827041552928, 5.68434164425938422133999472078, 6.39627690120702965574814482763, 7.28437006788105488837480646315, 8.104474742678596341240266451221, 9.270755005586392383106373521998, 10.00854369328378492884695847653