| L(s) = 1 | + (0.443 − 1.43i)2-s + (−1.53 − 0.601i)3-s + (−0.215 − 0.147i)4-s + (0.986 − 2.00i)5-s + (−1.54 + 1.93i)6-s + (2.62 + 0.320i)7-s + (2.04 − 1.63i)8-s + (−0.212 − 0.197i)9-s + (−2.44 − 2.30i)10-s + (−0.171 + 0.159i)11-s + (0.242 + 0.355i)12-s + (−5.60 + 1.27i)13-s + (1.62 − 3.63i)14-s + (−2.71 + 2.48i)15-s + (−1.62 − 4.14i)16-s + (−0.840 + 0.0629i)17-s + ⋯ |
| L(s) = 1 | + (0.313 − 1.01i)2-s + (−0.884 − 0.347i)3-s + (−0.107 − 0.0735i)4-s + (0.441 − 0.897i)5-s + (−0.630 + 0.790i)6-s + (0.992 + 0.121i)7-s + (0.722 − 0.576i)8-s + (−0.0709 − 0.0658i)9-s + (−0.773 − 0.729i)10-s + (−0.0516 + 0.0479i)11-s + (0.0698 + 0.102i)12-s + (−1.55 + 0.354i)13-s + (0.434 − 0.970i)14-s + (−0.701 + 0.640i)15-s + (−0.406 − 1.03i)16-s + (−0.203 + 0.0152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.582575 - 1.18528i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.582575 - 1.18528i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.986 + 2.00i)T \) |
| 7 | \( 1 + (-2.62 - 0.320i)T \) |
| good | 2 | \( 1 + (-0.443 + 1.43i)T + (-1.65 - 1.12i)T^{2} \) |
| 3 | \( 1 + (1.53 + 0.601i)T + (2.19 + 2.04i)T^{2} \) |
| 11 | \( 1 + (0.171 - 0.159i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (5.60 - 1.27i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (0.840 - 0.0629i)T + (16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-0.837 + 1.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.39 - 0.179i)T + (22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (-7.41 - 3.57i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (1.84 + 3.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.48 - 9.51i)T + (-13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (-2.71 - 3.40i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (3.89 + 3.11i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-1.90 + 6.16i)T + (-38.8 - 26.4i)T^{2} \) |
| 53 | \( 1 + (5.80 - 8.50i)T + (-19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-1.52 - 0.230i)T + (56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (8.68 - 5.92i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-1.24 + 0.716i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.03 + 4.35i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.67 - 5.43i)T + (-60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (0.731 - 1.26i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.0 - 2.74i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (6.28 + 5.83i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + 5.11iT - 97T^{2} \) |
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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
−11.92875537044641679439779590787, −11.15532780235494392239651927372, −10.12533420454403604475267815461, −9.081128333954845646685649611832, −7.76006150461935292440625999080, −6.60204648863239572465143563431, −5.13915293285485328221840825247, −4.58871522076040065339740617601, −2.55526286379301221401836096707, −1.18877030467927509142782579684,
2.37982495850736248939212418290, 4.65344217020543370909916415964, 5.34571739691463952417322417536, 6.26244855212108966039282546835, 7.28997308141608264622196909695, 8.060655228191162231615750402612, 9.781423697606639584281709245112, 10.76606799373708474468036530185, 11.18428514085158165180213436962, 12.31786651880356046339410667475
