| L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (1.93 − 1.11i)5-s + (−5.26 − 4.61i)7-s + 2.82·8-s + (−2.73 − 1.58i)10-s + (−5.41 + 9.37i)11-s − 19.2i·13-s + (−1.93 + 9.70i)14-s + (−2.00 − 3.46i)16-s + (8.89 + 5.13i)17-s + (−18.0 + 10.4i)19-s + 4.47i·20-s + 15.3·22-s + (10.5 + 18.2i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.387 − 0.223i)5-s + (−0.751 − 0.659i)7-s + 0.353·8-s + (−0.273 − 0.158i)10-s + (−0.491 + 0.852i)11-s − 1.48i·13-s + (−0.137 + 0.693i)14-s + (−0.125 − 0.216i)16-s + (0.523 + 0.302i)17-s + (−0.951 + 0.549i)19-s + 0.223i·20-s + 0.695·22-s + (0.458 + 0.793i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.09171792371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09171792371\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.93 + 1.11i)T \) |
| 7 | \( 1 + (5.26 + 4.61i)T \) |
| good | 11 | \( 1 + (5.41 - 9.37i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 19.2iT - 169T^{2} \) |
| 17 | \( 1 + (-8.89 - 5.13i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (18.0 - 10.4i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-10.5 - 18.2i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 19.0T + 841T^{2} \) |
| 31 | \( 1 + (34.6 + 20.0i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-25.1 - 43.6i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 22.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 48.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (57.6 - 33.2i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-2.47 + 4.28i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-24.4 - 14.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (60.6 - 35.0i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (9.65 - 16.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 49.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (115. + 66.4i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-45.0 - 78.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (34.3 - 19.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 68.6iT - 9.40e3T^{2} \) |
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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
−10.29290718359464454625075860096, −10.11244225900817924746265687285, −9.176827580196060652387065278059, −8.018379686054507854622942108198, −7.39810131272593004511629549251, −6.13927669010923904730159247527, −5.11745162229021769385546459348, −3.86478189623407463915381877812, −2.88753382905928832878244358810, −1.48824058073747597994895827917,
0.03705493961775167984934900181, 2.01776500905109291989489732740, 3.29127735642751247298499912992, 4.74601456701445777934509377686, 5.81871824958369754129854862843, 6.50143628203407963389247952366, 7.29573185067001452617800254565, 8.622049348852447715670507951916, 9.057553605166450479701638437339, 9.913791245087604126197668899120
