| L(s) = 1 | + (0.907 − 2.79i)2-s + (−0.510 − 0.370i)4-s + (−10.7 + 3.16i)5-s + 18.9·7-s + (17.5 − 12.7i)8-s + (−0.895 + 32.8i)10-s + (1.87 − 5.77i)11-s + (24.1 + 74.4i)13-s + (17.1 − 52.8i)14-s + (−21.2 − 65.2i)16-s + (31.0 − 22.5i)17-s + (75.2 − 54.6i)19-s + (6.64 + 2.36i)20-s + (−14.4 − 10.4i)22-s + (25.0 − 77.0i)23-s + ⋯ |
| L(s) = 1 | + (0.320 − 0.987i)2-s + (−0.0637 − 0.0463i)4-s + (−0.959 + 0.282i)5-s + 1.02·7-s + (0.774 − 0.562i)8-s + (−0.0283 + 1.03i)10-s + (0.0513 − 0.158i)11-s + (0.515 + 1.58i)13-s + (0.327 − 1.00i)14-s + (−0.331 − 1.02i)16-s + (0.443 − 0.322i)17-s + (0.908 − 0.659i)19-s + (0.0742 + 0.0263i)20-s + (−0.139 − 0.101i)22-s + (0.227 − 0.698i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.03940 - 1.19472i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.03940 - 1.19472i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (10.7 - 3.16i)T \) |
| good | 2 | \( 1 + (-0.907 + 2.79i)T + (-6.47 - 4.70i)T^{2} \) |
| 7 | \( 1 - 18.9T + 343T^{2} \) |
| 11 | \( 1 + (-1.87 + 5.77i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-24.1 - 74.4i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-31.0 + 22.5i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-75.2 + 54.6i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-25.0 + 77.0i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (9.02 + 6.55i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-181. + 131. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (33.2 + 102. i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-108. - 333. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 356.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (236. + 171. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-554. - 402. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-69.2 - 213. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (215. - 663. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-735. + 534. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-163. - 118. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (90.2 - 277. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (573. + 416. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (897. - 651. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (1.16 - 3.59i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (14.9 + 10.8i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−11.42208261745657368510078989142, −11.29212812044666599322290436772, −9.996953842594335378296411522291, −8.675494649550444513789332900083, −7.62512535182392980921463120796, −6.71732940623335846186206230180, −4.77743992538311031264769622735, −3.98104756400398044140919710630, −2.68999213930896531738862069663, −1.19028220252110415507946694306,
1.27155430307284946865386038624, 3.49192495884541054315654583500, 4.92869479642624319741501082013, 5.58584693583453696097168449987, 7.03575108553672153341910188534, 8.038747430628266542347547562110, 8.286452130604865916338675084303, 10.15725232815912673368400859362, 11.11191935974988945172210137296, 11.91555262752937115925684666472
