| L(s) = 1 | + (10.1 − 7.34i)2-s + (8.68 − 26.7i)4-s + (401. + 291. i)5-s + (534. − 1.64e3i)7-s + (385. + 1.18e3i)8-s + 6.20e3·10-s + (−1.62e3 + 4.10e3i)11-s + (−3.29e3 + 2.39e3i)13-s + (−6.67e3 − 2.05e4i)14-s + (1.55e4 + 1.12e4i)16-s + (1.88e4 + 1.36e4i)17-s + (−7.23e3 − 2.22e4i)19-s + (1.12e4 − 8.19e3i)20-s + (1.36e4 + 5.34e4i)22-s + 2.45e4·23-s + ⋯ |
| L(s) = 1 | + (0.893 − 0.649i)2-s + (0.0678 − 0.208i)4-s + (1.43 + 1.04i)5-s + (0.588 − 1.81i)7-s + (0.266 + 0.819i)8-s + 1.96·10-s + (−0.368 + 0.929i)11-s + (−0.416 + 0.302i)13-s + (−0.650 − 2.00i)14-s + (0.947 + 0.688i)16-s + (0.929 + 0.675i)17-s + (−0.241 − 0.744i)19-s + (0.315 − 0.229i)20-s + (0.274 + 1.06i)22-s + 0.421·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0869i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.40257 - 0.191846i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.40257 - 0.191846i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (1.62e3 - 4.10e3i)T \) |
| good | 2 | \( 1 + (-10.1 + 7.34i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (-401. - 291. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-534. + 1.64e3i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (3.29e3 - 2.39e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-1.88e4 - 1.36e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (7.23e3 + 2.22e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 2.45e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (2.72e4 - 8.40e4i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-1.68e5 + 1.22e5i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (-1.10e5 + 3.40e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (-1.53e5 - 4.73e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 + 5.45e4T + 2.71e11T^{2} \) |
| 47 | \( 1 + (7.34e4 + 2.26e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-7.82e5 + 5.68e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (4.61e4 - 1.42e5i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (1.35e6 + 9.81e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 4.38e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (7.52e5 + 5.46e5i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (6.98e5 - 2.15e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (2.83e6 - 2.06e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (2.01e6 + 1.46e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 3.39e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-5.76e6 + 4.18e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.84055023194893815702554712535, −11.26879906240801432946612783305, −10.51806762472577495182805112921, −9.811918733386978231871611273153, −7.73183753739880715838150841935, −6.75762094729904817308361539443, −5.19652034104118908583435012451, −4.06532271667868660940215855222, −2.66366369358411802027348848260, −1.53270824797967505923437000045,
1.18280196244276605291821780520, 2.66384268869057800835922890200, 4.93996560701534166504989755403, 5.51722992525615520586331564937, 6.11696091414820020219711195096, 8.167080016903035642753249509545, 9.165645372950603438669463838742, 10.12207868479077204317056556176, 11.97754292049302554505944142718, 12.71476581244139635132377959334
