| L(s) = 1 | + (−0.460 + 1.94i)2-s + (2.66 + 1.38i)3-s + (−3.57 − 1.79i)4-s + (−8.07 + 4.66i)5-s + (−3.91 + 4.54i)6-s + (4.91 + 2.83i)7-s + (5.13 − 6.13i)8-s + (5.18 + 7.35i)9-s + (−5.35 − 17.8i)10-s + (−1.85 + 3.21i)11-s + (−7.04 − 9.71i)12-s + (11.0 − 6.40i)13-s + (−7.78 + 8.25i)14-s + (−27.9 + 1.27i)15-s + (9.56 + 12.8i)16-s + 11.1·17-s + ⋯ |
| L(s) = 1 | + (−0.230 + 0.973i)2-s + (0.887 + 0.460i)3-s + (−0.893 − 0.448i)4-s + (−1.61 + 0.932i)5-s + (−0.652 + 0.757i)6-s + (0.701 + 0.405i)7-s + (0.642 − 0.766i)8-s + (0.576 + 0.817i)9-s + (−0.535 − 1.78i)10-s + (−0.168 + 0.292i)11-s + (−0.587 − 0.809i)12-s + (0.853 − 0.492i)13-s + (−0.555 + 0.589i)14-s + (−1.86 + 0.0848i)15-s + (0.597 + 0.801i)16-s + 0.657·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.710 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.424001 + 1.02991i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.424001 + 1.02991i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.460 - 1.94i)T \) |
| 3 | \( 1 + (-2.66 - 1.38i)T \) |
| good | 5 | \( 1 + (8.07 - 4.66i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-4.91 - 2.83i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (1.85 - 3.21i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-11.0 + 6.40i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 11.1T + 289T^{2} \) |
| 19 | \( 1 + 13.1T + 361T^{2} \) |
| 23 | \( 1 + (-20.2 + 11.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-14.6 - 8.47i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (3.32 - 1.91i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 13.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-3.05 - 5.29i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (11.3 - 19.7i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (49.8 + 28.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 60.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (55.3 + 95.8i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-73.2 - 42.3i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-16.0 - 27.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 38.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 13.6T + 5.32e3T^{2} \) |
| 79 | \( 1 + (4.14 + 2.39i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (2.70 - 4.68i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 98.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-42.1 + 73.0i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−14.90106314846745517353481294028, −14.41621850954970555224520297204, −12.88251112724313839034651516720, −11.22672530993766844723529296303, −10.21991177317469648214240063210, −8.486419219566000097346291199317, −8.057118607837778459158707756700, −6.88196295421224687688810429972, −4.78055576175253943080550624026, −3.45588250592526166792516550840,
1.14249688656434246858945883000, 3.51534335975260699601994114341, 4.53605300859357439826219246617, 7.57258736795819089923819880134, 8.306291380074103643840956797594, 9.098033685116035548902946682797, 10.90948628765739755737067973342, 11.83617916354123488029970122421, 12.73595540194645883941400721485, 13.69609643614642746681073335912
