| L(s) = 1 | + 2.82i·2-s + (−4.68 + 7.68i)3-s − 8.00·4-s − 31.2i·5-s + (−21.7 − 13.2i)6-s − 39.9·7-s − 22.6i·8-s + (−37.0 − 72.0i)9-s + 88.4·10-s + 110. i·11-s + (37.5 − 61.4i)12-s + 152.·13-s − 112. i·14-s + (240. + 146. i)15-s + 64.0·16-s − 99.8i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.520 + 0.853i)3-s − 0.500·4-s − 1.25i·5-s + (−0.603 − 0.368i)6-s − 0.814·7-s − 0.353i·8-s + (−0.457 − 0.889i)9-s + 0.884·10-s + 0.915i·11-s + (0.260 − 0.426i)12-s + 0.903·13-s − 0.576i·14-s + (1.06 + 0.651i)15-s + 0.250·16-s − 0.345i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.246027572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.246027572\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.82iT \) |
| 3 | \( 1 + (4.68 - 7.68i)T \) |
| 29 | \( 1 - 156. iT \) |
| good | 5 | \( 1 + 31.2iT - 625T^{2} \) |
| 7 | \( 1 + 39.9T + 2.40e3T^{2} \) |
| 11 | \( 1 - 110. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 152.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 99.8iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 426.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 238. iT - 2.79e5T^{2} \) |
| 31 | \( 1 + 60.5T + 9.23e5T^{2} \) |
| 37 | \( 1 - 176.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.44e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.00e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.25e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.45e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.67e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 7.05e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.09e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 554. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.07e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 8.09e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.28e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.29e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 7.71e3T + 8.85e7T^{2} \) |
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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
−12.37467152345784469134606295119, −11.23127777207711056055997300449, −9.686730793638133519494600506355, −9.427450644784566507361969883100, −8.229834835932077464278909999921, −6.77883332734274668107161819266, −5.60415438304168677213309581201, −4.78412812740981834185295992199, −3.61862153731053047417984352422, −0.805266362487852077853896041347,
0.816885803912991189805134409193, 2.58349203174739086920628319464, 3.58726440367464546541172530891, 5.70132650000075722585536506077, 6.49763810560790125810254402725, 7.60072803407060836071797027873, 8.900045125341604747569147520009, 10.31516467961719659517771543776, 10.96916997237671121321673177546, 11.74322295475263440517537861588
