| L(s) = 1 | + (−6.47 + 4.70i)2-s + (6.77 + 4.92i)3-s + (19.7 − 60.8i)4-s + 35.5·5-s − 67.0·6-s + (−136. + 419. i)7-s + (158. + 486. i)8-s + (−654. − 2.01e3i)9-s + (−230. + 167. i)10-s + (−320. + 985. i)11-s + (433. − 315. i)12-s + (6.31e3 + 4.58e3i)13-s + (−1.09e3 − 3.35e3i)14-s + (241. + 175. i)15-s + (−3.31e3 − 2.40e3i)16-s + (6.66e3 + 2.05e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.144 + 0.105i)3-s + (0.154 − 0.475i)4-s + 0.127·5-s − 0.126·6-s + (−0.150 + 0.462i)7-s + (0.109 + 0.336i)8-s + (−0.299 − 0.920i)9-s + (−0.0728 + 0.0529i)10-s + (−0.0725 + 0.223i)11-s + (0.0724 − 0.0526i)12-s + (0.797 + 0.579i)13-s + (−0.106 − 0.327i)14-s + (0.0184 + 0.0134i)15-s + (−0.202 − 0.146i)16-s + (0.328 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.143451 + 0.682190i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.143451 + 0.682190i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.47 - 4.70i)T \) |
| 31 | \( 1 + (1.12e5 - 1.22e5i)T \) |
| good | 3 | \( 1 + (-6.77 - 4.92i)T + (675. + 2.07e3i)T^{2} \) |
| 5 | \( 1 - 35.5T + 7.81e4T^{2} \) |
| 7 | \( 1 + (136. - 419. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 11 | \( 1 + (320. - 985. i)T + (-1.57e7 - 1.14e7i)T^{2} \) |
| 13 | \( 1 + (-6.31e3 - 4.58e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-6.66e3 - 2.05e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (1.20e4 - 8.77e3i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 + (1.03e4 + 3.17e4i)T + (-2.75e9 + 2.00e9i)T^{2} \) |
| 29 | \( 1 + (1.75e5 - 1.27e5i)T + (5.33e9 - 1.64e10i)T^{2} \) |
| 37 | \( 1 + 2.40e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (4.80e5 - 3.48e5i)T + (6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 + (1.68e5 - 1.22e5i)T + (8.39e10 - 2.58e11i)T^{2} \) |
| 47 | \( 1 + (2.01e5 + 1.46e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (1.49e4 + 4.60e4i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-1.61e5 - 1.17e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 - 2.87e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.57e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-1.77e6 - 5.47e6i)T + (-7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-3.42e5 + 1.05e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (1.68e6 + 5.17e6i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-2.48e6 + 1.80e6i)T + (8.38e12 - 2.58e13i)T^{2} \) |
| 89 | \( 1 + (9.56e5 - 2.94e6i)T + (-3.57e13 - 2.59e13i)T^{2} \) |
| 97 | \( 1 + (4.91e4 - 1.51e5i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−14.36748151781013166426840352478, −12.88180943746289258823086581315, −11.68776670888343612016614858014, −10.36463305553886271399859968417, −9.182894290112141438151696182158, −8.358551183343982161398742012497, −6.72579437014689221035101304048, −5.69692390789734652790312332135, −3.68400588367938957771177786382, −1.67596414363576209430846158706,
0.29286536745287942271555441473, 2.04125554414660120575365255443, 3.61807278119437181457952210807, 5.54565367673082757048949915138, 7.31516083809025352750561147243, 8.310505067911759382008081342768, 9.629211633789341595256234344764, 10.74269389845454443910652094621, 11.63956868438378468801657655667, 13.22375344574339946731655953027
