| L(s) = 1 | + (0.243 + 0.304i)2-s + (7.00 + 1.05i)3-s + (1.74 − 7.65i)4-s + (−2.70 − 1.84i)5-s + (1.38 + 2.39i)6-s + (−4.58 + 7.93i)7-s + (5.56 − 2.68i)8-s + (22.2 + 6.84i)9-s + (−0.0953 − 1.27i)10-s + (7.09 + 31.0i)11-s + (20.3 − 51.7i)12-s + (−0.518 + 6.91i)13-s + (−3.53 + 0.532i)14-s + (−17.0 − 15.7i)15-s + (−54.3 − 26.1i)16-s + (−19.6 + 13.3i)17-s + ⋯ |
| L(s) = 1 | + (0.0859 + 0.107i)2-s + (1.34 + 0.203i)3-s + (0.218 − 0.956i)4-s + (−0.241 − 0.164i)5-s + (0.0940 + 0.162i)6-s + (−0.247 + 0.428i)7-s + (0.246 − 0.118i)8-s + (0.822 + 0.253i)9-s + (−0.00301 − 0.0402i)10-s + (0.194 + 0.852i)11-s + (0.488 − 1.24i)12-s + (−0.0110 + 0.147i)13-s + (−0.0674 + 0.0101i)14-s + (−0.292 − 0.271i)15-s + (−0.849 − 0.409i)16-s + (−0.279 + 0.190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.89854 - 0.133913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89854 - 0.133913i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (20.2 - 281. i)T \) |
| good | 2 | \( 1 + (-0.243 - 0.304i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (-7.00 - 1.05i)T + (25.8 + 7.95i)T^{2} \) |
| 5 | \( 1 + (2.70 + 1.84i)T + (45.6 + 116. i)T^{2} \) |
| 7 | \( 1 + (4.58 - 7.93i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-7.09 - 31.0i)T + (-1.19e3 + 577. i)T^{2} \) |
| 13 | \( 1 + (0.518 - 6.91i)T + (-2.17e3 - 327. i)T^{2} \) |
| 17 | \( 1 + (19.6 - 13.3i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (70.3 - 21.6i)T + (5.66e3 - 3.86e3i)T^{2} \) |
| 23 | \( 1 + (66.8 - 62.0i)T + (909. - 1.21e4i)T^{2} \) |
| 29 | \( 1 + (-281. + 42.4i)T + (2.33e4 - 7.18e3i)T^{2} \) |
| 31 | \( 1 + (-45.5 + 116. i)T + (-2.18e4 - 2.02e4i)T^{2} \) |
| 37 | \( 1 + (10.4 + 18.1i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-125. - 157. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 47 | \( 1 + (-70.7 + 309. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (16.7 + 224. i)T + (-1.47e5 + 2.21e4i)T^{2} \) |
| 59 | \( 1 + (-681. - 328. i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (284. + 725. i)T + (-1.66e5 + 1.54e5i)T^{2} \) |
| 67 | \( 1 + (753. - 232. i)T + (2.48e5 - 1.69e5i)T^{2} \) |
| 71 | \( 1 + (-293. - 272. i)T + (2.67e4 + 3.56e5i)T^{2} \) |
| 73 | \( 1 + (-28.1 + 375. i)T + (-3.84e5 - 5.79e4i)T^{2} \) |
| 79 | \( 1 + (-662. + 1.14e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-1.20e3 - 182. i)T + (5.46e5 + 1.68e5i)T^{2} \) |
| 89 | \( 1 + (375. + 56.5i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 + (-24.2 - 106. i)T + (-8.22e5 + 3.95e5i)T^{2} \) |
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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
−15.21002429162668330887212581637, −14.51022738256906811904387623903, −13.47104175012946279862557373125, −11.97847998961645319080621012557, −10.23250678650053026673734487485, −9.307527351447682296962795916828, −8.057401473342956853507374872470, −6.35561185548272102826810294902, −4.36163493485643604617953160452, −2.26166959414386884679928700829,
2.71626539577334430630870304873, 3.91145901196375743071873723230, 6.85701301581421569146323931257, 8.095574619540351586380065847824, 8.895551811800141226915402218597, 10.71709342596370390231746644277, 12.16691837134538575632374935810, 13.38104113375440043859825782317, 14.07507352359941156400229025169, 15.43208787532934580581474148323
