| L(s) = 1 | + (6.49 + 4.72i)2-s + (−3.46 − 10.6i)3-s + (−19.6 − 60.3i)4-s + (182. + 211. i)5-s + (27.8 − 85.5i)6-s + 756.·7-s + (475. − 1.46e3i)8-s + (1.66e3 − 1.21e3i)9-s + (187. + 2.23e3i)10-s + (3.21e3 + 2.33e3i)11-s + (−575. + 417. i)12-s + (−458. + 332. i)13-s + (4.91e3 + 3.56e3i)14-s + (1.62e3 − 2.67e3i)15-s + (3.42e3 − 2.48e3i)16-s + (−2.81e3 + 8.67e3i)17-s + ⋯ |
| L(s) = 1 | + (0.574 + 0.417i)2-s + (−0.0740 − 0.227i)3-s + (−0.153 − 0.471i)4-s + (0.653 + 0.757i)5-s + (0.0525 − 0.161i)6-s + 0.833·7-s + (0.328 − 1.01i)8-s + (0.762 − 0.554i)9-s + (0.0592 + 0.707i)10-s + (0.728 + 0.529i)11-s + (−0.0960 + 0.0697i)12-s + (−0.0578 + 0.0420i)13-s + (0.478 + 0.347i)14-s + (0.124 − 0.204i)15-s + (0.209 − 0.151i)16-s + (−0.139 + 0.428i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0413i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.55582 + 0.0528808i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.55582 + 0.0528808i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-182. - 211. i)T \) |
| good | 2 | \( 1 + (-6.49 - 4.72i)T + (39.5 + 121. i)T^{2} \) |
| 3 | \( 1 + (3.46 + 10.6i)T + (-1.76e3 + 1.28e3i)T^{2} \) |
| 7 | \( 1 - 756.T + 8.23e5T^{2} \) |
| 11 | \( 1 + (-3.21e3 - 2.33e3i)T + (6.02e6 + 1.85e7i)T^{2} \) |
| 13 | \( 1 + (458. - 332. i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (2.81e3 - 8.67e3i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-3.34e3 + 1.03e4i)T + (-7.23e8 - 5.25e8i)T^{2} \) |
| 23 | \( 1 + (5.00e4 + 3.63e4i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (4.93e4 + 1.51e5i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (7.41e4 - 2.28e5i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (2.60e5 - 1.89e5i)T + (2.93e10 - 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-4.11e5 + 2.98e5i)T + (6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 - 5.63e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.35e5 - 7.26e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-1.11e5 - 3.42e5i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (1.80e6 - 1.30e6i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + (1.04e6 + 7.61e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + (9.03e5 - 2.77e6i)T + (-4.90e12 - 3.56e12i)T^{2} \) |
| 71 | \( 1 + (1.37e6 + 4.22e6i)T + (-7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-3.82e6 - 2.77e6i)T + (3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (4.17e5 + 1.28e6i)T + (-1.55e13 + 1.12e13i)T^{2} \) |
| 83 | \( 1 + (5.80e5 - 1.78e6i)T + (-2.19e13 - 1.59e13i)T^{2} \) |
| 89 | \( 1 + (4.97e6 + 3.61e6i)T + (1.36e13 + 4.20e13i)T^{2} \) |
| 97 | \( 1 + (9.14e5 + 2.81e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−15.56717854166906506326459991514, −14.62363539399813503456489147663, −13.84456821179229592024293907241, −12.39771481047908299498027802336, −10.66859037595849208624114690318, −9.462050264839139261527600955799, −7.15551398101803935444508695259, −6.04599096433326858515091393147, −4.31222792644101072338843184478, −1.56884053953656365546621994257,
1.78395759588188625767947939555, 4.12004852235065430106311540299, 5.36741743749653942507934276741, 7.84433561162246848196626234133, 9.271360329314094483857376679621, 11.01543879837415753634726554313, 12.25893494689721741745714383837, 13.41660317166317323406000667317, 14.31257567904518521909487496289, 16.20390219413034600300650542296
