L(s) = 1 | − 32i·2-s − 1.02e3·4-s + (−237 − 3.11e3i)5-s + 3.27e4i·8-s − 5.90e4·9-s + (−9.97e4 + 7.58e3i)10-s + 2.91e5i·13-s + 1.04e6·16-s + 1.81e6i·17-s + 1.88e6i·18-s + (2.42e5 + 3.19e6i)20-s + (−9.65e6 + 1.47e6i)25-s + 9.32e6·26-s − 3.23e7·29-s − 3.35e7i·32-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + (−0.0758 − 0.997i)5-s + i·8-s − 0.999·9-s + (−0.997 + 0.0758i)10-s + 0.784i·13-s + 16-s + 1.27i·17-s + 0.999i·18-s + (0.0758 + 0.997i)20-s + (−0.988 + 0.151i)25-s + 0.784·26-s − 1.57·29-s − i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0758 - 0.997i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.0758 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(0.0163215 + 0.0176101i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0163215 + 0.0176101i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 32iT \) |
| 5 | \( 1 + (237 + 3.11e3i)T \) |
good | 3 | \( 1 + 5.90e4T^{2} \) |
| 7 | \( 1 + 2.82e8T^{2} \) |
| 11 | \( 1 - 2.59e10T^{2} \) |
| 13 | \( 1 - 2.91e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 - 1.81e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 6.13e12T^{2} \) |
| 23 | \( 1 + 4.14e13T^{2} \) |
| 29 | \( 1 + 3.23e7T + 4.20e14T^{2} \) |
| 31 | \( 1 - 8.19e14T^{2} \) |
| 37 | \( 1 + 1.38e8iT - 4.80e15T^{2} \) |
| 41 | \( 1 + 2.07e8T + 1.34e16T^{2} \) |
| 43 | \( 1 + 2.16e16T^{2} \) |
| 47 | \( 1 + 5.25e16T^{2} \) |
| 53 | \( 1 - 2.93e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 5.11e17T^{2} \) |
| 61 | \( 1 - 1.33e9T + 7.13e17T^{2} \) |
| 67 | \( 1 + 1.82e18T^{2} \) |
| 71 | \( 1 - 3.25e18T^{2} \) |
| 73 | \( 1 + 1.78e9iT - 4.29e18T^{2} \) |
| 79 | \( 1 - 9.46e18T^{2} \) |
| 83 | \( 1 + 1.55e19T^{2} \) |
| 89 | \( 1 + 8.56e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 1.48e10iT - 7.37e19T^{2} \) |
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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
−14.66048857301158082226361315217, −13.33683898830623415046039873339, −12.20674054119528787219085390849, −11.05668457554543724731985048082, −9.348816326607864177592522580533, −8.322223659141558793924353590275, −5.51166735838906271496744679882, −3.88620322910430489482607886180, −1.80879552975791149161111937351, −0.01010640148670726492398611891,
3.19718105483879006788069383338, 5.39170558775144095392667094485, 6.84170185753842172088754329020, 8.200229849834407582880479684846, 9.847991418686205980398288620904, 11.49265046338919852755875581141, 13.42479577316980359220625371892, 14.51370641964613354730091130475, 15.43458927908061322101577835386, 16.84809582052003103040373979644