| L(s) = 1 | + (25.0 − 9.99i)3-s − 55.9i·5-s + 134.·7-s + (529. − 501. i)9-s − 1.15e3i·11-s − 78.2·13-s + (−558. − 1.40e3i)15-s − 54.6i·17-s + 1.98e3·19-s + (3.37e3 − 1.34e3i)21-s − 1.69e4i·23-s − 3.12e3·25-s + (8.27e3 − 1.78e4i)27-s − 1.84e4i·29-s + 3.83e4·31-s + ⋯ |
| L(s) = 1 | + (0.929 − 0.370i)3-s − 0.447i·5-s + 0.392·7-s + (0.726 − 0.687i)9-s − 0.867i·11-s − 0.0356·13-s + (−0.165 − 0.415i)15-s − 0.0111i·17-s + 0.289·19-s + (0.364 − 0.145i)21-s − 1.39i·23-s − 0.199·25-s + (0.420 − 0.907i)27-s − 0.757i·29-s + 1.28·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.929i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.370 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.08146 - 1.41145i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08146 - 1.41145i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-25.0 + 9.99i)T \) |
| 5 | \( 1 + 55.9iT \) |
| good | 7 | \( 1 - 134.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 1.15e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 78.2T + 4.82e6T^{2} \) |
| 17 | \( 1 + 54.6iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 1.98e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.69e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.84e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.83e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 2.12e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 8.89e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 6.55e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.63e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 9.71e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.86e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 1.19e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 1.42e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 3.63e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.76e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 6.27e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 9.91e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + 8.27e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 8.48e5T + 8.32e11T^{2} \) |
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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
−13.69211044649068857982840663921, −12.70842588203687498224656906675, −11.49493716953340080381707333422, −9.941715988261936103295692173159, −8.673538886806545065618976679146, −7.902709268900341598781574256028, −6.31016220653244606112891834815, −4.45005385886818797764771121472, −2.77493473199104799414377705936, −1.03012574108023921802639793223,
1.91586137548899324560593878744, 3.48120084481491599835236972921, 4.97999795328231671524406894411, 7.02210870918937926815661213888, 8.104270837869228818275038075719, 9.446876693206452197706021354535, 10.38591027394160929663096171877, 11.75401953496415035977267873190, 13.19148625095062466492921268321, 14.19669675542397104652152178686
