| L(s) = 1 | + (0.173 + 0.984i)2-s + (−0.915 − 1.47i)3-s + (−0.939 + 0.342i)4-s + (−0.192 − 0.161i)5-s + (1.28 − 1.15i)6-s + (0.939 + 0.342i)7-s + (−0.5 − 0.866i)8-s + (−1.32 + 2.69i)9-s + (0.125 − 0.218i)10-s + (3.36 − 2.82i)11-s + (1.36 + 1.06i)12-s + (0.135 − 0.769i)13-s + (−0.173 + 0.984i)14-s + (−0.0614 + 0.431i)15-s + (0.766 − 0.642i)16-s + (3.45 − 5.98i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.528 − 0.848i)3-s + (−0.469 + 0.171i)4-s + (−0.0862 − 0.0724i)5-s + (0.526 − 0.472i)6-s + (0.355 + 0.129i)7-s + (−0.176 − 0.306i)8-s + (−0.441 + 0.897i)9-s + (0.0398 − 0.0689i)10-s + (1.01 − 0.851i)11-s + (0.393 + 0.308i)12-s + (0.0376 − 0.213i)13-s + (−0.0464 + 0.263i)14-s + (−0.0158 + 0.111i)15-s + (0.191 − 0.160i)16-s + (0.837 − 1.45i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12852 - 0.312538i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12852 - 0.312538i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.915 + 1.47i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| good | 5 | \( 1 + (0.192 + 0.161i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-3.36 + 2.82i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.135 + 0.769i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.45 + 5.98i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.33 + 4.05i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.91 + 2.15i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.28 - 7.26i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.28 - 0.833i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (4.96 - 8.60i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.838 + 4.75i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.13 + 2.63i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.46 - 0.898i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 8.77T + 53T^{2} \) |
| 59 | \( 1 + (-9.74 - 8.18i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (7.12 + 2.59i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (2.62 - 14.9i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-3.23 + 5.59i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.16 - 3.74i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.486 - 2.75i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.375 + 2.13i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (7.64 + 13.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.57 + 1.32i)T + (16.8 - 95.5i)T^{2} \) |
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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
−11.49463293705719118054611054289, −10.56195363569569212667083518635, −9.036856931289623968125258540703, −8.446229741705050476523483428084, −7.24058201098928121266585764244, −6.66986448345096994787521056668, −5.55995322507495590138056520027, −4.72167275435228304581134423668, −2.98258239034110672864427212349, −0.927559485796711003879517422332,
1.59843885827870950347669333055, 3.59482506239294343150013343750, 4.23063528859828237813067432850, 5.41223376454690242275384547639, 6.42364734486417124599711038128, 7.84745553023028717889065943187, 9.095074778441570223775704915119, 9.748412762317364925805710268922, 10.66094018382948959522419677530, 11.31173994932498516639406350447
