L(s) = 1 | + (−3.57 + 2.06i)2-s + (4.50 − 7.79i)4-s + 3.05i·5-s + (−5.78 − 3.34i)7-s + 4.12i·8-s + (−6.28 − 10.8i)10-s + (27.9 − 16.1i)11-s + (−22.1 − 41.3i)13-s + 27.5·14-s + (27.4 + 47.6i)16-s + (14.4 − 24.9i)17-s + (87.6 + 50.5i)19-s + (23.7 + 13.7i)20-s + (−66.4 + 115. i)22-s + (59.4 + 103. i)23-s + ⋯ |
L(s) = 1 | + (−1.26 + 0.728i)2-s + (0.562 − 0.974i)4-s + 0.272i·5-s + (−0.312 − 0.180i)7-s + 0.182i·8-s + (−0.198 − 0.344i)10-s + (0.765 − 0.441i)11-s + (−0.472 − 0.881i)13-s + 0.526·14-s + (0.429 + 0.744i)16-s + (0.205 − 0.356i)17-s + (1.05 + 0.610i)19-s + (0.265 + 0.153i)20-s + (−0.644 + 1.11i)22-s + (0.539 + 0.934i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.782014 + 0.190975i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.782014 + 0.190975i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (22.1 + 41.3i)T \) |
good | 2 | \( 1 + (3.57 - 2.06i)T + (4 - 6.92i)T^{2} \) |
| 5 | \( 1 - 3.05iT - 125T^{2} \) |
| 7 | \( 1 + (5.78 + 3.34i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-27.9 + 16.1i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (-14.4 + 24.9i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-87.6 - 50.5i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-59.4 - 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-80.0 - 138. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 38.0iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-283. + 163. i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (48.5 - 28.0i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-63.9 + 110. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 517. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 695.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (568. + 328. i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (350. - 607. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (49.5 - 28.5i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-267. - 154. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 - 389. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 901.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 687. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (-927. + 535. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.51e3 + 877. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−13.20577273449940885582999036497, −11.93138198436358746994662213739, −10.62949163342451828854632225461, −9.741474704147710934502419325378, −8.844316081680791063959505050449, −7.65262821125048351548745010225, −6.85983210228863351597068601705, −5.55774146400745512084857634167, −3.38575986463969649093693677978, −0.881693975055655711825576983157,
1.08648428413736721756562809407, 2.70912744466107263606092106516, 4.67947398457206658774981201108, 6.56357460656310603197347596007, 7.84799364439858443385826512311, 9.131305719330767331360972877237, 9.538500867301798025514737542315, 10.76088981496111298117279945107, 11.78165489959414372743459058855, 12.49775994281198609275225026326