| L(s) = 1 | + (0.967 − 2.65i)2-s + (−0.716 + 8.97i)3-s + (−6.12 − 5.14i)4-s + (−20.4 − 3.59i)5-s + (23.1 + 10.5i)6-s + (−31.5 + 26.4i)7-s + (−19.5 + 11.3i)8-s + (−79.9 − 12.8i)9-s + (−29.3 + 50.7i)10-s + (−75.0 + 13.2i)11-s + (50.5 − 51.2i)12-s + (−76.2 + 27.7i)13-s + (39.8 + 109. i)14-s + (46.9 − 180. i)15-s + (11.1 + 63.0i)16-s + (−9.50 − 5.48i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (−0.0796 + 0.996i)3-s + (−0.383 − 0.321i)4-s + (−0.816 − 0.143i)5-s + (0.643 + 0.293i)6-s + (−0.643 + 0.540i)7-s + (−0.306 + 0.176i)8-s + (−0.987 − 0.158i)9-s + (−0.293 + 0.507i)10-s + (−0.620 + 0.109i)11-s + (0.350 − 0.356i)12-s + (−0.451 + 0.164i)13-s + (0.203 + 0.558i)14-s + (0.208 − 0.802i)15-s + (0.0434 + 0.246i)16-s + (−0.0328 − 0.0189i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.149000 + 0.405299i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149000 + 0.405299i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.967 + 2.65i)T \) |
| 3 | \( 1 + (0.716 - 8.97i)T \) |
| good | 5 | \( 1 + (20.4 + 3.59i)T + (587. + 213. i)T^{2} \) |
| 7 | \( 1 + (31.5 - 26.4i)T + (416. - 2.36e3i)T^{2} \) |
| 11 | \( 1 + (75.0 - 13.2i)T + (1.37e4 - 5.00e3i)T^{2} \) |
| 13 | \( 1 + (76.2 - 27.7i)T + (2.18e4 - 1.83e4i)T^{2} \) |
| 17 | \( 1 + (9.50 + 5.48i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-332. - 576. i)T + (-6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-222. + 265. i)T + (-4.85e4 - 2.75e5i)T^{2} \) |
| 29 | \( 1 + (-386. + 1.06e3i)T + (-5.41e5 - 4.54e5i)T^{2} \) |
| 31 | \( 1 + (563. + 473. i)T + (1.60e5 + 9.09e5i)T^{2} \) |
| 37 | \( 1 + (1.21e3 - 2.11e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-98.9 - 271. i)T + (-2.16e6 + 1.81e6i)T^{2} \) |
| 43 | \( 1 + (50.8 + 288. i)T + (-3.21e6 + 1.16e6i)T^{2} \) |
| 47 | \( 1 + (-2.02e3 - 2.40e3i)T + (-8.47e5 + 4.80e6i)T^{2} \) |
| 53 | \( 1 + 113. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (3.36e3 + 593. i)T + (1.13e7 + 4.14e6i)T^{2} \) |
| 61 | \( 1 + (3.60e3 - 3.02e3i)T + (2.40e6 - 1.36e7i)T^{2} \) |
| 67 | \( 1 + (-469. + 170. i)T + (1.54e7 - 1.29e7i)T^{2} \) |
| 71 | \( 1 + (6.67e3 + 3.85e3i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-4.41e3 - 7.65e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (1.02e3 + 372. i)T + (2.98e7 + 2.50e7i)T^{2} \) |
| 83 | \( 1 + (-259. + 713. i)T + (-3.63e7 - 3.05e7i)T^{2} \) |
| 89 | \( 1 + (1.06e4 - 6.15e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (1.13e3 + 6.45e3i)T + (-8.31e7 + 3.02e7i)T^{2} \) |
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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
−15.19937057941672146921134339990, −13.95203570224799659358987353305, −12.40115878247788820437724476019, −11.68705911324387339262191654361, −10.34959943018787620944877626804, −9.455678541221295712835478972098, −8.051643451054861812842327946769, −5.77521723269449769319491579657, −4.35333696995517890739202884164, −3.00690637872147322414689041901,
0.22904101469723885785034034778, 3.22347130073752417078119761832, 5.27905177533758323708971857409, 7.00026047814944396150599045112, 7.50199585968449447098740206317, 8.987448845210437751622237307010, 10.88255885835642551562922877097, 12.16519905961770193356098854119, 13.14953940957441782474477627710, 14.01205320005845089932846406271
