| L(s) = 1 | + (−6.55 − 9.22i)2-s + (−42.1 + 120. i)4-s + 328. i·5-s + 343i·7-s + (1.39e3 − 403. i)8-s + (3.02e3 − 2.15e3i)10-s − 3.62e3·11-s − 7.66e3·13-s + (3.16e3 − 2.24e3i)14-s + (−1.28e4 − 1.01e4i)16-s + 1.44e4i·17-s + 2.75e4i·19-s + (−3.96e4 − 1.38e4i)20-s + (2.37e4 + 3.34e4i)22-s − 5.37e4·23-s + ⋯ |
| L(s) = 1 | + (−0.579 − 0.815i)2-s + (−0.329 + 0.944i)4-s + 1.17i·5-s + 0.377i·7-s + (0.960 − 0.278i)8-s + (0.957 − 0.680i)10-s − 0.821·11-s − 0.967·13-s + (0.308 − 0.218i)14-s + (−0.783 − 0.621i)16-s + 0.711i·17-s + 0.921i·19-s + (−1.10 − 0.386i)20-s + (0.475 + 0.669i)22-s − 0.921·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.580i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1302572957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1302572957\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.55 + 9.22i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 343iT \) |
| good | 5 | \( 1 - 328. iT - 7.81e4T^{2} \) |
| 11 | \( 1 + 3.62e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.66e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.44e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 - 2.75e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 5.37e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.69e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 1.67e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 - 1.63e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.51e5iT - 1.94e11T^{2} \) |
| 43 | \( 1 - 2.92e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 6.12e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.12e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 + 1.31e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.01e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.50e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 - 1.43e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.70e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.64e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 - 5.60e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.13e6iT - 4.42e13T^{2} \) |
| 97 | \( 1 + 9.17e6T + 8.07e13T^{2} \) |
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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
−11.13536929352862049393365629328, −10.38742948271233253355903865733, −9.812868653705121842167117478303, −8.488598588777262321145274183642, −7.64571457644813062805673185518, −6.65473328324476691364911733140, −5.18289372086354811126665001412, −3.67225292332445643507777888873, −2.73314750329466831704748225735, −1.79193911367542064890516525890,
0.04795695451568097083647789127, 0.807795176517807525612909178302, 2.32098653675676295060774039169, 4.47873808073497521348711254869, 5.06406444484429874711001728279, 6.23192332871580892030444055970, 7.55534252473577444166803233583, 8.074081259474325599676634099221, 9.303271660456985389165964474033, 9.800354870582034803680233155146
