| L(s) = 1 | + (1.53 + 1.28i)2-s + (4.60 − 2.40i)3-s + (0.694 + 3.93i)4-s + (3.45 + 1.25i)5-s + (10.1 + 2.23i)6-s + (0.157 − 0.890i)7-s + (−4.00 + 6.92i)8-s + (15.4 − 22.1i)9-s + (3.67 + 6.35i)10-s + (−0.906 + 0.329i)11-s + (12.6 + 16.4i)12-s + (−34.0 + 28.5i)13-s + (1.38 − 1.16i)14-s + (18.9 − 2.52i)15-s + (−15.0 + 5.47i)16-s + (−23.7 − 41.1i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.886 − 0.463i)3-s + (0.0868 + 0.492i)4-s + (0.308 + 0.112i)5-s + (0.690 + 0.151i)6-s + (0.00847 − 0.0480i)7-s + (−0.176 + 0.306i)8-s + (0.570 − 0.821i)9-s + (0.116 + 0.201i)10-s + (−0.0248 + 0.00904i)11-s + (0.305 + 0.396i)12-s + (−0.725 + 0.608i)13-s + (0.0264 − 0.0221i)14-s + (0.325 − 0.0434i)15-s + (−0.234 + 0.0855i)16-s + (−0.339 − 0.587i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.26422 + 0.388202i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.26422 + 0.388202i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.53 - 1.28i)T \) |
| 3 | \( 1 + (-4.60 + 2.40i)T \) |
| good | 5 | \( 1 + (-3.45 - 1.25i)T + (95.7 + 80.3i)T^{2} \) |
| 7 | \( 1 + (-0.157 + 0.890i)T + (-322. - 117. i)T^{2} \) |
| 11 | \( 1 + (0.906 - 0.329i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (34.0 - 28.5i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (23.7 + 41.1i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (40.7 - 70.5i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (22.7 + 129. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (108. + 90.9i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-33.1 - 188. i)T + (-2.79e4 + 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-172. - 299. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-268. + 224. i)T + (1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (51.5 - 18.7i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (65.7 - 373. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 - 583.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-180. - 65.5i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-94.1 + 533. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (543. - 455. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-484. - 839. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-253. + 439. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (716. + 601. i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (704. + 590. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (-17.2 + 29.8i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-509. + 185. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−14.58334052009955215452942298164, −14.02266104324183353750770655202, −12.89769923700892335811739011442, −11.89681733822260956363400735985, −10.01270432968691680286143048849, −8.689280775068626637677196204600, −7.44600856196412190818655623250, −6.27221641898458823460952728064, −4.28944642371534367163098812115, −2.43065799921087386887447250565,
2.31222167541048651743001807908, 3.96368763032532789867021871813, 5.49145535830841276375571706445, 7.51437304481678979285857049225, 9.075668120949572205249362492725, 10.05983727832844868015360941764, 11.27075032082651044887363991315, 12.88031187395021060012235233994, 13.54520502109920655200976460286, 14.84452192352044638844197177328
