| L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (−1.93 − 1.11i)5-s + (4.24 + 5.56i)7-s + 2.82·8-s + (2.73 − 1.58i)10-s + (−5.42 − 9.40i)11-s − 0.772i·13-s + (−9.81 + 1.26i)14-s + (−2.00 + 3.46i)16-s + (−16.7 + 9.68i)17-s + (22.5 + 13.0i)19-s + 4.47i·20-s + 15.3·22-s + (−6.84 + 11.8i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.387 − 0.223i)5-s + (0.606 + 0.795i)7-s + 0.353·8-s + (0.273 − 0.158i)10-s + (−0.493 − 0.854i)11-s − 0.0593i·13-s + (−0.701 + 0.0902i)14-s + (−0.125 + 0.216i)16-s + (−0.986 + 0.569i)17-s + (1.18 + 0.686i)19-s + 0.223i·20-s + 0.698·22-s + (−0.297 + 0.515i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8004863545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8004863545\) |
| \(L(2)\) |
|
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 | \( 1 + (-4.24 - 5.56i)T \) |
| good | 11 | \( 1 + (5.42 + 9.40i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 0.772iT - 169T^{2} \) |
| 17 | \( 1 + (16.7 - 9.68i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-22.5 - 13.0i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (6.84 - 11.8i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 6.99T + 841T^{2} \) |
| 31 | \( 1 + (-22.7 + 13.1i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (32.3 - 55.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 5.54iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 68.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-19.5 - 11.3i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-37.2 - 64.5i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (96.6 - 55.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (46.9 + 27.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-22.1 - 38.3i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 31.9T + 5.04e3T^{2} \) |
| 73 | \( 1 + (92.6 - 53.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (14.8 - 25.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 15.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-31.8 - 18.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 134. iT - 9.40e3T^{2} \) |
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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
−10.70109986082001255942633733430, −9.752292904024850809126146781181, −8.669061700800281688374711437716, −8.281283933298675982503976451177, −7.41056010653076702794336566425, −6.17168769670502002700056984082, −5.44727518515358783292295572986, −4.47182409515382352565772335076, −3.03933182091519835321563723565, −1.43662112740278258604207721188,
0.34360222982663745598767098377, 1.89533387947697529947186186754, 3.13381477646299696173270969337, 4.37600763614322817127574260208, 5.04306255296452563952190654016, 6.83324299499872712309712801227, 7.41220498143729357607614382922, 8.299753337494582751073728652697, 9.274720616025770033332249627215, 10.20469712551408641145269008629
