| L(s) = 1 | + (−0.259 − 0.508i)2-s + (0.838 + 5.29i)3-s + (2.15 − 2.97i)4-s + (−3.73 − 3.32i)5-s + (2.47 − 1.79i)6-s + (1.66 − 1.66i)7-s + (−4.32 − 0.685i)8-s + (−18.7 + 6.09i)9-s + (−0.724 + 2.76i)10-s + (0.984 − 3.02i)11-s + (17.5 + 8.94i)12-s + (1.52 − 3.00i)13-s + (−1.27 − 0.414i)14-s + (14.4 − 22.5i)15-s + (−3.76 − 11.5i)16-s + (−2.34 + 14.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.129 − 0.254i)2-s + (0.279 + 1.76i)3-s + (0.539 − 0.743i)4-s + (−0.746 − 0.665i)5-s + (0.412 − 0.299i)6-s + (0.237 − 0.237i)7-s + (−0.541 − 0.0857i)8-s + (−2.08 + 0.677i)9-s + (−0.0724 + 0.276i)10-s + (0.0894 − 0.275i)11-s + (1.46 + 0.745i)12-s + (0.117 − 0.230i)13-s + (−0.0911 − 0.0296i)14-s + (0.965 − 1.50i)15-s + (−0.235 − 0.724i)16-s + (−0.137 + 0.869i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.914845 + 0.169521i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.914845 + 0.169521i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (3.73 + 3.32i)T \) |
| good | 2 | \( 1 + (0.259 + 0.508i)T + (-2.35 + 3.23i)T^{2} \) |
| 3 | \( 1 + (-0.838 - 5.29i)T + (-8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (-1.66 + 1.66i)T - 49iT^{2} \) |
| 11 | \( 1 + (-0.984 + 3.02i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (-1.52 + 3.00i)T + (-99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (2.34 - 14.7i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-13.4 - 18.5i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (13.8 - 7.04i)T + (310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-10.2 + 14.1i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-9.99 + 7.25i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-0.734 - 0.373i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (8.67 + 26.7i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (42.9 + 42.9i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-46.4 + 7.34i)T + (2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (0.616 + 3.89i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-77.9 + 25.3i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-15.8 + 48.7i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-11.9 + 75.3i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-76.2 - 55.3i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (101. - 51.5i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (47.6 - 65.6i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-68.6 - 10.8i)T + (6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (33.4 + 10.8i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-4.45 + 0.705i)T + (8.94e3 - 2.90e3i)T^{2} \) |
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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
−17.00220438349511852264715183018, −15.92854892807496740964262716767, −15.32785479620744146128573703619, −14.18170209941814146718697479075, −11.83620797045898702595751049969, −10.72647457273992502234395117389, −9.723599696420477027562014881655, −8.329302941153338822005442376376, −5.46419068157988654446933675120, −3.83355975003627897744198810923,
2.74885216966418794152214705996, 6.66081862542100131176156565699, 7.42103830341100601788438132572, 8.519293910065167974980853494833, 11.49800322540929296285759227246, 12.07069510386075968658512467034, 13.43091884764077174760655272028, 14.74582716221486166377770890154, 16.13426490551208514013979157018, 17.77269721662134697504269848025
