| L(s) = 1 | + (−1.30 − 1.51i)2-s + (−4.75 − 3.05i)3-s + (−0.569 + 3.95i)4-s + (−6.73 + 14.7i)5-s + (1.60 + 11.1i)6-s + (22.9 + 6.72i)7-s + (6.73 − 4.32i)8-s + (2.04 + 4.48i)9-s + (31.1 − 9.13i)10-s + (−43.2 + 49.9i)11-s + (14.8 − 17.0i)12-s + (−66.3 + 19.4i)13-s + (−19.8 − 43.4i)14-s + (77.1 − 49.5i)15-s + (−15.3 − 4.50i)16-s + (2.87 + 20.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (−0.914 − 0.587i)3-s + (−0.0711 + 0.494i)4-s + (−0.602 + 1.31i)5-s + (0.109 + 0.761i)6-s + (1.23 + 0.363i)7-s + (0.297 − 0.191i)8-s + (0.0758 + 0.166i)9-s + (0.984 − 0.288i)10-s + (−1.18 + 1.36i)11-s + (0.356 − 0.410i)12-s + (−1.41 + 0.415i)13-s + (−0.378 − 0.829i)14-s + (1.32 − 0.852i)15-s + (−0.239 − 0.0704i)16-s + (0.0410 + 0.285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0300 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0300 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.316830 + 0.307443i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.316830 + 0.307443i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-68.6 - 86.3i)T \) |
| good | 3 | \( 1 + (4.75 + 3.05i)T + (11.2 + 24.5i)T^{2} \) |
| 5 | \( 1 + (6.73 - 14.7i)T + (-81.8 - 94.4i)T^{2} \) |
| 7 | \( 1 + (-22.9 - 6.72i)T + (288. + 185. i)T^{2} \) |
| 11 | \( 1 + (43.2 - 49.9i)T + (-189. - 1.31e3i)T^{2} \) |
| 13 | \( 1 + (66.3 - 19.4i)T + (1.84e3 - 1.18e3i)T^{2} \) |
| 17 | \( 1 + (-2.87 - 20.0i)T + (-4.71e3 + 1.38e3i)T^{2} \) |
| 19 | \( 1 + (-9.51 + 66.1i)T + (-6.58e3 - 1.93e3i)T^{2} \) |
| 29 | \( 1 + (21.4 + 149. i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-55.7 + 35.8i)T + (1.23e4 - 2.70e4i)T^{2} \) |
| 37 | \( 1 + (46.8 + 102. i)T + (-3.31e4 + 3.82e4i)T^{2} \) |
| 41 | \( 1 + (105. - 230. i)T + (-4.51e4 - 5.20e4i)T^{2} \) |
| 43 | \( 1 + (-221. - 142. i)T + (3.30e4 + 7.23e4i)T^{2} \) |
| 47 | \( 1 + 290.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (34.6 + 10.1i)T + (1.25e5 + 8.04e4i)T^{2} \) |
| 59 | \( 1 + (-110. + 32.5i)T + (1.72e5 - 1.11e5i)T^{2} \) |
| 61 | \( 1 + (-230. + 148. i)T + (9.42e4 - 2.06e5i)T^{2} \) |
| 67 | \( 1 + (-476. - 550. i)T + (-4.28e4 + 2.97e5i)T^{2} \) |
| 71 | \( 1 + (-116. - 134. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (89.6 - 623. i)T + (-3.73e5 - 1.09e5i)T^{2} \) |
| 79 | \( 1 + (-636. + 186. i)T + (4.14e5 - 2.66e5i)T^{2} \) |
| 83 | \( 1 + (33.4 + 73.1i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (704. + 452. i)T + (2.92e5 + 6.41e5i)T^{2} \) |
| 97 | \( 1 + (533. - 1.16e3i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
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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
−15.36436027363582814131791043778, −14.65298331715128006312773116876, −12.77676503902424399275192819159, −11.67372085352648130685755894220, −11.17393353693859695441921296084, −9.878775151286544845102491812743, −7.74783280751103645433963712465, −7.04222547187434132091737666018, −4.93171914893608530673872369116, −2.36334347254133467450214743501,
0.43856390928630386584503191069, 4.85680711756941711112055199798, 5.32382357949041402824691206794, 7.76202137822028028750697109046, 8.540149987674828164290598595359, 10.32224268307751011261921736463, 11.23643187732995020873153119636, 12.45915857795366815119935472339, 14.05428985025145771477455338354, 15.43660851400682076728669751686
