| L(s) = 1 | + (3.23 − 2.35i)2-s + (4.94 − 15.2i)4-s + (−37.6 + 51.7i)5-s + (−37.4 − 12.1i)7-s + (−19.7 − 60.8i)8-s + 255. i·10-s + (377. + 136. i)11-s + (−458. − 631. i)13-s + (−149. + 48.6i)14-s + (−207. − 150. i)16-s + (445. + 323. i)17-s + (1.03e3 − 336. i)19-s + (601. + 828. i)20-s + (1.54e3 − 444. i)22-s − 3.74e3i·23-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.154 − 0.475i)4-s + (−0.672 + 0.926i)5-s + (−0.288 − 0.0938i)7-s + (−0.109 − 0.336i)8-s + 0.809i·10-s + (0.940 + 0.340i)11-s + (−0.752 − 1.03i)13-s + (−0.204 + 0.0663i)14-s + (−0.202 − 0.146i)16-s + (0.374 + 0.271i)17-s + (0.657 − 0.213i)19-s + (0.336 + 0.463i)20-s + (0.679 − 0.195i)22-s − 1.47i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0489 + 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0489 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.036605166\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.036605166\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 + 2.35i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-377. - 136. i)T \) |
| good | 5 | \( 1 + (37.6 - 51.7i)T + (-965. - 2.97e3i)T^{2} \) |
| 7 | \( 1 + (37.4 + 12.1i)T + (1.35e4 + 9.87e3i)T^{2} \) |
| 13 | \( 1 + (458. + 631. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-445. - 323. i)T + (4.38e5 + 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.03e3 + 336. i)T + (2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + 3.74e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-2.77e3 + 8.54e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-4.60e3 + 3.34e3i)T + (8.84e6 - 2.72e7i)T^{2} \) |
| 37 | \( 1 + (3.09e3 - 9.52e3i)T + (-5.61e7 - 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-1.23e3 - 3.80e3i)T + (-9.37e7 + 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.36e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.02e4 + 3.32e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.65e4 + 2.27e4i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-3.00e4 - 9.77e3i)T + (5.78e8 + 4.20e8i)T^{2} \) |
| 61 | \( 1 + (1.09e4 - 1.51e4i)T + (-2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + 1.18e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-1.53e4 + 2.11e4i)T + (-5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.53e4 - 5.00e3i)T + (1.67e9 + 1.21e9i)T^{2} \) |
| 79 | \( 1 + (2.49e4 + 3.43e4i)T + (-9.50e8 + 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-3.02e4 - 2.19e4i)T + (1.21e9 + 3.74e9i)T^{2} \) |
| 89 | \( 1 + 1.15e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (8.29e4 - 6.02e4i)T + (2.65e9 - 8.16e9i)T^{2} \) |
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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
−11.61246346863070284214109143120, −10.39309414218716284868772880034, −9.810256900022765149409383717588, −8.185139035545858914813852524852, −7.08159601442704891893457379925, −6.17610370223502721660674192612, −4.67163095289585583104254331353, −3.53002825191402912000527030676, −2.52965710793875427344946910287, −0.57356925104337542764344837499,
1.25951577838207519473936156022, 3.27614128947903425049138097818, 4.38383216447976005876287078325, 5.35239569247702098439300237703, 6.69425679262801951276215270422, 7.63873148965709107315824407240, 8.824114577596918541528961826205, 9.548905231691593898250992946463, 11.25312670690010603635933351639, 12.12002683590884839940297821375
