| L(s) = 1 | + (−1.30 + 1.51i)2-s + (7.02 − 4.51i)3-s + (−0.569 − 3.95i)4-s + (−7.76 − 17.0i)5-s + (−2.37 + 16.5i)6-s + (−9.77 + 2.86i)7-s + (6.73 + 4.32i)8-s + (17.7 − 38.9i)9-s + (35.8 + 10.5i)10-s + (27.1 + 31.2i)11-s + (−21.8 − 25.2i)12-s + (59.4 + 17.4i)13-s + (8.46 − 18.5i)14-s + (−131. − 84.4i)15-s + (−15.3 + 4.50i)16-s + (−4.50 + 31.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 + 0.534i)2-s + (1.35 − 0.869i)3-s + (−0.0711 − 0.494i)4-s + (−0.694 − 1.52i)5-s + (−0.161 + 1.12i)6-s + (−0.527 + 0.154i)7-s + (0.297 + 0.191i)8-s + (0.658 − 1.44i)9-s + (1.13 + 0.333i)10-s + (0.743 + 0.857i)11-s + (−0.526 − 0.607i)12-s + (1.26 + 0.372i)13-s + (0.161 − 0.353i)14-s + (−2.26 − 1.45i)15-s + (−0.239 + 0.0704i)16-s + (−0.0642 + 0.447i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.647 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.28098 - 0.592061i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28098 - 0.592061i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.30 - 1.51i)T \) |
| 23 | \( 1 + (0.295 + 110. i)T \) |
| good | 3 | \( 1 + (-7.02 + 4.51i)T + (11.2 - 24.5i)T^{2} \) |
| 5 | \( 1 + (7.76 + 17.0i)T + (-81.8 + 94.4i)T^{2} \) |
| 7 | \( 1 + (9.77 - 2.86i)T + (288. - 185. i)T^{2} \) |
| 11 | \( 1 + (-27.1 - 31.2i)T + (-189. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-59.4 - 17.4i)T + (1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (4.50 - 31.3i)T + (-4.71e3 - 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-12.1 - 84.4i)T + (-6.58e3 + 1.93e3i)T^{2} \) |
| 29 | \( 1 + (-37.7 + 262. i)T + (-2.34e4 - 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-64.2 - 41.2i)T + (1.23e4 + 2.70e4i)T^{2} \) |
| 37 | \( 1 + (80.9 - 177. i)T + (-3.31e4 - 3.82e4i)T^{2} \) |
| 41 | \( 1 + (56.9 + 124. i)T + (-4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (235. - 151. i)T + (3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 - 313.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (566. - 166. i)T + (1.25e5 - 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-203. - 59.8i)T + (1.72e5 + 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-194. - 125. i)T + (9.42e4 + 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-429. + 496. i)T + (-4.28e4 - 2.97e5i)T^{2} \) |
| 71 | \( 1 + (373. - 430. i)T + (-5.09e4 - 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-54.2 - 377. i)T + (-3.73e5 + 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-154. - 45.4i)T + (4.14e5 + 2.66e5i)T^{2} \) |
| 83 | \( 1 + (-230. + 504. i)T + (-3.74e5 - 4.32e5i)T^{2} \) |
| 89 | \( 1 + (669. - 430. i)T + (2.92e5 - 6.41e5i)T^{2} \) |
| 97 | \( 1 + (108. + 236. i)T + (-5.97e5 + 6.89e5i)T^{2} \) |
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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
−15.22129610512160633158157127580, −13.96766797304870161049477831740, −12.92512965247395723009462107124, −12.05317542151255232523812572611, −9.592399179127371567381167393274, −8.598086457349142935361720783980, −8.022469818899630046771166940562, −6.47740710594075567132168430177, −4.08843451551474484558362287393, −1.40679391629051829761043428226,
3.12381100418240651416098739774, 3.62165581062397095836982180030, 6.91072913300577111729008955220, 8.336181689236611522421819537313, 9.382285187349410580784751283168, 10.62357241821330984548581730845, 11.37950639317563205711302456202, 13.48641087563235345876579227280, 14.31783123107932665256414618624, 15.47384980704328581701054726264
