| L(s) = 1 | + (−4.84 − 2.79i)2-s + (1.66 + 4.92i)3-s + (11.6 + 20.1i)4-s + (11.6 + 3.12i)5-s + (5.72 − 28.4i)6-s + (14.4 − 3.87i)7-s − 85.3i·8-s + (−21.4 + 16.3i)9-s + (−47.7 − 47.7i)10-s + (52.0 − 13.9i)11-s + (−79.8 + 90.7i)12-s + (16.2 + 28.0i)13-s + (−80.8 − 21.6i)14-s + (3.97 + 62.5i)15-s + (−145. + 252. i)16-s + (29.2 − 63.6i)17-s + ⋯ |
| L(s) = 1 | + (−1.71 − 0.988i)2-s + (0.319 + 0.947i)3-s + (1.45 + 2.51i)4-s + (1.04 + 0.279i)5-s + (0.389 − 1.93i)6-s + (0.780 − 0.209i)7-s − 3.77i·8-s + (−0.795 + 0.605i)9-s + (−1.50 − 1.50i)10-s + (1.42 − 0.382i)11-s + (−1.92 + 2.18i)12-s + (0.345 + 0.598i)13-s + (−1.54 − 0.413i)14-s + (0.0684 + 1.07i)15-s + (−2.27 + 3.93i)16-s + (0.417 − 0.908i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.183i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 - 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.15844 + 0.107261i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15844 + 0.107261i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.66 - 4.92i)T \) |
| 17 | \( 1 + (-29.2 + 63.6i)T \) |
| good | 2 | \( 1 + (4.84 + 2.79i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-11.6 - 3.12i)T + (108. + 62.5i)T^{2} \) |
| 7 | \( 1 + (-14.4 + 3.87i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-52.0 + 13.9i)T + (1.15e3 - 665.5i)T^{2} \) |
| 13 | \( 1 + (-16.2 - 28.0i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 19 | \( 1 + 17.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (6.58 - 24.5i)T + (-1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (36.7 + 137. i)T + (-2.11e4 + 1.21e4i)T^{2} \) |
| 31 | \( 1 + (-136. - 36.5i)T + (2.57e4 + 1.48e4i)T^{2} \) |
| 37 | \( 1 + (-21.2 + 21.2i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (108. - 405. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-310. - 179. i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-144. + 249. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 - 170. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (403. - 233. i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-274. + 73.5i)T + (1.96e5 - 1.13e5i)T^{2} \) |
| 67 | \( 1 + (11.5 + 19.9i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (345. - 345. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-189. + 189. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (445. - 119. i)T + (4.26e5 - 2.46e5i)T^{2} \) |
| 83 | \( 1 + (692. + 400. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + 310.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-333. - 1.24e3i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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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
−11.72425010420685970900319067768, −11.33405456975135364286109848683, −10.25187225415930223029172327307, −9.500782375913074520446168100676, −8.933118712569755082783457379128, −7.82346053876626809539343537992, −6.38104215314193417210023148853, −4.12875823260136816273718768391, −2.72778721898838027594409567347, −1.37825995576998155768146109600,
1.18531101222400407818086477506, 1.92761561234196651233631890009, 5.58748987998252079672421565808, 6.30576772374944909307641183977, 7.36771531695508794577221000425, 8.423382237582699193723830841906, 9.013861361767115442990712157959, 10.01234031487864511557520815146, 11.16718251981034528989889100167, 12.34510376574855482922928746699
