| L(s) = 1 | + (0.699 − 1.87i)2-s + (−1.09 + 1.33i)3-s + (−3.02 − 2.62i)4-s + (0.535 + 1.76i)5-s + (1.74 + 2.99i)6-s + (0.225 − 1.13i)7-s + (−7.02 + 3.82i)8-s + (−0.585 − 2.94i)9-s + (3.68 + 0.230i)10-s + (17.6 + 1.74i)11-s + (6.82 − 1.16i)12-s + (4.38 + 1.33i)13-s + (−1.96 − 1.21i)14-s + (−2.95 − 1.22i)15-s + (2.26 + 15.8i)16-s + (−5.46 − 13.1i)17-s + ⋯ |
| L(s) = 1 | + (0.349 − 0.936i)2-s + (−0.366 + 0.446i)3-s + (−0.755 − 0.655i)4-s + (0.107 + 0.352i)5-s + (0.290 + 0.499i)6-s + (0.0322 − 0.162i)7-s + (−0.877 + 0.478i)8-s + (−0.0650 − 0.326i)9-s + (0.368 + 0.0230i)10-s + (1.60 + 0.158i)11-s + (0.569 − 0.0972i)12-s + (0.337 + 0.102i)13-s + (−0.140 − 0.0868i)14-s + (−0.196 − 0.0814i)15-s + (0.141 + 0.989i)16-s + (−0.321 − 0.776i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.34947 - 1.10395i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.34947 - 1.10395i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.699 + 1.87i)T \) |
| 3 | \( 1 + (1.09 - 1.33i)T \) |
| good | 5 | \( 1 + (-0.535 - 1.76i)T + (-20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-0.225 + 1.13i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-17.6 - 1.74i)T + (118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-4.38 - 1.33i)T + (140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (5.46 + 13.1i)T + (-204. + 204. i)T^{2} \) |
| 19 | \( 1 + (-14.4 + 7.74i)T + (200. - 300. i)T^{2} \) |
| 23 | \( 1 + (30.3 + 20.2i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-33.2 + 3.27i)T + (824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-24.7 + 24.7i)T - 961iT^{2} \) |
| 37 | \( 1 + (19.4 + 10.4i)T + (760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-46.7 - 31.2i)T + (643. + 1.55e3i)T^{2} \) |
| 43 | \( 1 + (37.3 + 45.5i)T + (-360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-3.34 - 8.06i)T + (-1.56e3 + 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-73.3 - 7.22i)T + (2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (30.9 + 101. i)T + (-2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (68.5 - 83.5i)T + (-725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (12.9 + 10.5i)T + (875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-72.8 - 14.4i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (123. - 24.5i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (35.4 - 85.5i)T + (-4.41e3 - 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-35.5 - 66.4i)T + (-3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (-84.7 - 126. i)T + (-3.03e3 + 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-28.7 - 28.7i)T + 9.40e3iT^{2} \) |
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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
−11.00517797250359276733130429520, −10.13949372116832958637722035908, −9.412120719841829723390336945644, −8.537497067896291115270883578192, −6.81089383297739513647652351820, −6.00554238480889148049715323313, −4.64826234527872405066690191455, −3.93123173644136679900217319938, −2.59734569014974331914686102317, −0.900948743559955004569505610463,
1.28011423484891326110577800635, 3.48139079633813579494250862620, 4.58337392284183456566412802787, 5.80731017497916023765627275607, 6.40439935081522159887623428710, 7.41078761454328215238063306035, 8.486907500769116315226439034986, 9.124247563627227716066085471999, 10.32513203761733006470714699416, 11.90953995334229347007404019688
