L(s) = 1 | + (−0.342 + 0.939i)2-s + (−1.85 + 0.326i)3-s + (−0.766 − 0.642i)4-s + (0.326 − 1.85i)6-s + (−2.65 − 1.53i)7-s + (0.866 − 0.500i)8-s + (0.5 − 0.181i)9-s + (2.17 + 3.76i)11-s + (1.62 + 0.939i)12-s + (−5.67 − i)13-s + (2.34 − 1.96i)14-s + (0.173 + 0.984i)16-s + (−0.705 + 1.93i)17-s + 0.532i·18-s + (−4.34 + 0.405i)19-s + ⋯ |
L(s) = 1 | + (−0.241 + 0.664i)2-s + (−1.06 + 0.188i)3-s + (−0.383 − 0.321i)4-s + (0.133 − 0.755i)6-s + (−1.00 − 0.579i)7-s + (0.306 − 0.176i)8-s + (0.166 − 0.0606i)9-s + (0.655 + 1.13i)11-s + (0.469 + 0.271i)12-s + (−1.57 − 0.277i)13-s + (0.627 − 0.526i)14-s + (0.0434 + 0.246i)16-s + (−0.171 + 0.470i)17-s + 0.125i·18-s + (−0.995 + 0.0929i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.541011 + 0.0937709i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.541011 + 0.0937709i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.342 - 0.939i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (4.34 - 0.405i)T \) |
good | 3 | \( 1 + (1.85 - 0.326i)T + (2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (2.65 + 1.53i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.17 - 3.76i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.67 + i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.705 - 1.93i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-4.98 + 5.94i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.87 + 1.41i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.22 + 7.32i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4iT - 37T^{2} \) |
| 41 | \( 1 + (0.0248 + 0.140i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.99 - 8.34i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.01 - 5.53i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.25 + 3.87i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-7.19 - 2.61i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (6.94 + 5.82i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.15 - 5.92i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.34 + 3.64i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.07 - 0.365i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.0641 - 0.363i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.652 - 0.376i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.308 - 1.75i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-3.65 + 10.0i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−10.03674552403315541297233507373, −9.497633621613033416033268137917, −8.335953687427637787642685276529, −7.26175819270105615208070704650, −6.59608154398219978627057122542, −6.05918459990072334688999329997, −4.73214029617625270178859529582, −4.36768958701052574975791649002, −2.57649801935901285190809377758, −0.51919524426680490637262646038,
0.72919425099920595018028078923, 2.50576784411544634074070045644, 3.43745460249120346777691906311, 4.82338006792951156612405635777, 5.64623213583314435063340687347, 6.55156608632297108738912170378, 7.23784597254247101446255389305, 8.785296152611945936216186137187, 9.117700132607252365818211626358, 10.15024816284682731298639088179