L(s) = 1 | + (−1.93 − 1.11i)2-s + (0.5 + 0.866i)4-s + (−1.93 + 1.11i)5-s + (3 − 5.19i)7-s + 6.70i·8-s + 5.00·10-s + (3.87 + 2.23i)11-s + (−8 − 13.8i)13-s + (−11.6 + 6.70i)14-s + (9.5 − 16.4i)16-s − 4.47i·17-s − 2·19-s + (−1.93 − 1.11i)20-s + (−5 − 8.66i)22-s + (−11.6 + 6.70i)23-s + ⋯ |
L(s) = 1 | + (−0.968 − 0.559i)2-s + (0.125 + 0.216i)4-s + (−0.387 + 0.223i)5-s + (0.428 − 0.742i)7-s + 0.838i·8-s + 0.500·10-s + (0.352 + 0.203i)11-s + (−0.615 − 1.06i)13-s + (−0.829 + 0.479i)14-s + (0.593 − 1.02i)16-s − 0.263i·17-s − 0.105·19-s + (−0.0968 − 0.0559i)20-s + (−0.227 − 0.393i)22-s + (−0.505 + 0.291i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0524834 + 0.144197i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0524834 + 0.144197i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.93 - 1.11i)T \) |
good | 2 | \( 1 + (1.93 + 1.11i)T + (2 + 3.46i)T^{2} \) |
| 7 | \( 1 + (-3 + 5.19i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.87 - 2.23i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (8 + 13.8i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 4.47iT - 289T^{2} \) |
| 19 | \( 1 + 2T + 361T^{2} \) |
| 23 | \( 1 + (11.6 - 6.70i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (27.1 + 15.6i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-9 - 15.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 16T + 1.36e3T^{2} \) |
| 41 | \( 1 + (54.2 - 31.3i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (8 - 13.8i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (42.6 + 24.5i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 4.47iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (3.87 - 2.23i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (41 - 71.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (12 + 20.7i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74T + 5.32e3T^{2} \) |
| 79 | \( 1 + (69 - 119. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (81.3 + 46.9i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-83 + 143. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.25598700486923443960283562808, −9.903480085102080214606449592352, −8.668005366602920796262544304131, −7.87711990774809287071946415055, −7.10925933311462358363762477834, −5.57687945345266415416219105165, −4.44513144876836018399967967255, −3.01380016694650550325465066719, −1.51832174230599798408075286416, −0.094264536159551397494271509950,
1.80995418471933533059442319523, 3.66750442383478570843400769290, 4.84292658764540987784707405111, 6.20089988989085521849222319231, 7.14215922317609484793223257465, 8.052884517060027916364915472510, 8.831056060001859962071768715916, 9.394646913540634051056978869999, 10.47515963175351484911534819110, 11.67844394722651102010894893778