| L(s) = 1 | + (3.76 − 2.73i)2-s + (4.22 − 12.9i)4-s + (−5.34 − 9.82i)5-s − 26.0·7-s + (−8.14 − 25.0i)8-s + (−46.9 − 22.3i)10-s + (2.03 − 1.47i)11-s + (−32.0 − 23.2i)13-s + (−98.0 + 71.2i)14-s + (−10.7 − 7.84i)16-s + (26.2 + 80.8i)17-s + (−44.1 − 135. i)19-s + (−150. + 27.9i)20-s + (3.61 − 11.1i)22-s + (127. − 92.3i)23-s + ⋯ |
| L(s) = 1 | + (1.33 − 0.967i)2-s + (0.527 − 1.62i)4-s + (−0.477 − 0.878i)5-s − 1.40·7-s + (−0.359 − 1.10i)8-s + (−1.48 − 0.707i)10-s + (0.0557 − 0.0405i)11-s + (−0.683 − 0.496i)13-s + (−1.87 + 1.36i)14-s + (−0.168 − 0.122i)16-s + (0.374 + 1.15i)17-s + (−0.532 − 1.64i)19-s + (−1.67 + 0.312i)20-s + (0.0350 − 0.107i)22-s + (1.15 − 0.837i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.143879 + 2.13001i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143879 + 2.13001i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (5.34 + 9.82i)T \) |
| good | 2 | \( 1 + (-3.76 + 2.73i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 + 26.0T + 343T^{2} \) |
| 11 | \( 1 + (-2.03 + 1.47i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (32.0 + 23.2i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-26.2 - 80.8i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (44.1 + 135. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-127. + 92.3i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-30.9 + 95.3i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (53.5 + 164. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-48.0 - 34.9i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (287. + 208. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 109.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-17.9 + 55.2i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-120. + 371. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-333. - 242. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (290. - 210. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-108. - 333. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (52.7 - 162. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-754. + 548. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (405. - 1.24e3i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-202. - 621. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (-857. + 622. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-198. + 610. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−11.59845357578244707236989665793, −10.59408287313128618534850988518, −9.634077684860090240978843595173, −8.491505223805623759524921109534, −6.87947687366736973607613340608, −5.69503066729667957689426321065, −4.63870399913403465995568983486, −3.64269159428250190384982956019, −2.51944739381819692101130512902, −0.55060459015254465766994528209,
2.99713579375233251360407797753, 3.69398992555173655866878260353, 5.07829453762690129069119112227, 6.30098025097428303202268665408, 6.94412911952356344499819909607, 7.71424846367843145743516501902, 9.390027602605916733715214890072, 10.40783432062149037001779390717, 11.83854214234099060094197425789, 12.45037533099074824574699959413
