| L(s) = 1 | + (−7.82 + 5.68i)2-s + (−10.6 + 32.7i)4-s + (100. + 73.3i)5-s + (−177. + 545. i)7-s + (−485. − 1.49e3i)8-s − 1.20e3·10-s + (2.51e3 − 3.63e3i)11-s + (−2.55e3 + 1.85e3i)13-s + (−1.71e3 − 5.27e3i)14-s + (8.73e3 + 6.34e3i)16-s + (−1.50e4 − 1.09e4i)17-s + (756. + 2.32e3i)19-s + (−3.47e3 + 2.52e3i)20-s + (992. + 4.26e4i)22-s − 5.35e4·23-s + ⋯ |
| L(s) = 1 | + (−0.691 + 0.502i)2-s + (−0.0830 + 0.255i)4-s + (0.361 + 0.262i)5-s + (−0.195 + 0.600i)7-s + (−0.335 − 1.03i)8-s − 0.381·10-s + (0.568 − 0.822i)11-s + (−0.323 + 0.234i)13-s + (−0.166 − 0.513i)14-s + (0.533 + 0.387i)16-s + (−0.741 − 0.538i)17-s + (0.0252 + 0.0778i)19-s + (−0.0970 + 0.0704i)20-s + (0.0198 + 0.854i)22-s − 0.918·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.950 + 0.309i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.891106 - 0.141386i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.891106 - 0.141386i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (-2.51e3 + 3.63e3i)T \) |
| good | 2 | \( 1 + (7.82 - 5.68i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-100. - 73.3i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (177. - 545. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (2.55e3 - 1.85e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (1.50e4 + 1.09e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-756. - 2.32e3i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + 5.35e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-1.54e4 + 4.74e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (3.30e3 - 2.40e3i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.26e5 + 3.90e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (-1.47e5 - 4.53e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 3.84e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (8.52e4 + 2.62e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-1.01e6 + 7.39e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-6.96e5 + 2.14e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-2.38e6 - 1.73e6i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 3.06e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (7.19e5 + 5.22e5i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (-1.34e6 + 4.12e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (8.52e5 - 6.19e5i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (5.72e6 + 4.15e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 5.68e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (8.00e6 - 5.81e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.40411888908682037283706167815, −11.40093808196453098333998379139, −9.907197613716327698657121869035, −9.078217228427679679299140849576, −8.146134832727938191961396711138, −6.84198636328447156753991666676, −5.90247036263432433689029720187, −4.01037872499656695518047418908, −2.47445203325320933661789233830, −0.41979212298278992253220151774,
1.05090711369211955288851589570, 2.22253218568454580033691566165, 4.19456282841800284016902750717, 5.60742859187842608795272999394, 7.01270551011057099288770708058, 8.453729504512605491797025884867, 9.540548177505371840437004174560, 10.19513063844663375054470134933, 11.25762472285271552851065522041, 12.41927775437359746846525815290
