| L(s) = 1 | + (−1.45 + 1.36i)2-s + (2.96 − 0.459i)3-s + (0.260 − 3.99i)4-s + (0.868 + 4.92i)5-s + (−3.69 + 4.72i)6-s + (−1.57 − 1.31i)7-s + (5.07 + 6.18i)8-s + (8.57 − 2.72i)9-s + (−8.00 − 6.00i)10-s + (−2.15 + 12.2i)11-s + (−1.06 − 11.9i)12-s + (−1.86 + 5.13i)13-s + (4.09 − 0.224i)14-s + (4.84 + 14.2i)15-s + (−15.8 − 2.08i)16-s + (14.0 − 8.11i)17-s + ⋯ |
| L(s) = 1 | + (−0.729 + 0.683i)2-s + (0.988 − 0.153i)3-s + (0.0652 − 0.997i)4-s + (0.173 + 0.985i)5-s + (−0.616 + 0.787i)6-s + (−0.224 − 0.188i)7-s + (0.634 + 0.772i)8-s + (0.952 − 0.302i)9-s + (−0.800 − 0.600i)10-s + (−0.195 + 1.10i)11-s + (−0.0884 − 0.996i)12-s + (−0.143 + 0.394i)13-s + (0.292 − 0.0160i)14-s + (0.322 + 0.947i)15-s + (−0.991 − 0.130i)16-s + (0.826 − 0.477i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.13630 + 0.996683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13630 + 0.996683i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.45 - 1.36i)T \) |
| 3 | \( 1 + (-2.96 + 0.459i)T \) |
| good | 5 | \( 1 + (-0.868 - 4.92i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (1.57 + 1.31i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (2.15 - 12.2i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (1.86 - 5.13i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-14.0 + 8.11i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-15.7 - 9.07i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-9.99 - 11.9i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (6.36 - 2.31i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (29.5 - 24.8i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (10.0 - 5.80i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (8.64 - 23.7i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (23.2 + 4.09i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (23.4 - 27.8i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 41.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (17.5 + 99.2i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-57.2 + 68.2i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-17.9 + 49.2i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-82.8 + 47.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-53.2 + 92.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (97.1 - 35.3i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (120. - 43.9i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-110. - 63.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (0.630 - 3.57i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.39614922745561062302677635916, −10.99510515546067633502017872126, −9.813739138146494100154804290897, −9.613594597667485515556423986698, −8.170579809676991115215024581010, −7.22707333700255639983867206050, −6.76873278733907994834143630191, −5.09361866273006682955360836698, −3.28927643717071180130970000465, −1.79481242594074360374704564559,
1.06529872760290664119699865137, 2.73116062719810901902492209972, 3.83376976183693103109270003062, 5.38407835581747859686515276426, 7.28089462446807315988619930802, 8.328664621611169862153795952734, 8.886278132487307698954682536149, 9.732391178822257338978996592167, 10.67139679064672181828997238745, 11.88435699442761070406858007792
