| L(s) = 1 | + (1.00 − 0.296i)2-s + (0.654 + 0.755i)3-s + (−0.751 + 0.483i)4-s + (0.142 + 0.989i)5-s + (0.884 + 0.568i)6-s + (−0.00344 + 0.00753i)7-s + (−1.99 + 2.30i)8-s + (−0.142 + 0.989i)9-s + (0.436 + 0.956i)10-s + (3.44 + 1.01i)11-s + (−0.857 − 0.251i)12-s + (1.04 + 2.27i)13-s + (−0.00123 + 0.00862i)14-s + (−0.654 + 0.755i)15-s + (−0.587 + 1.28i)16-s + (1.60 + 1.02i)17-s + ⋯ |
| L(s) = 1 | + (0.713 − 0.209i)2-s + (0.378 + 0.436i)3-s + (−0.375 + 0.241i)4-s + (0.0636 + 0.442i)5-s + (0.361 + 0.232i)6-s + (−0.00130 + 0.00284i)7-s + (−0.704 + 0.813i)8-s + (−0.0474 + 0.329i)9-s + (0.138 + 0.302i)10-s + (1.03 + 0.304i)11-s + (−0.247 − 0.0726i)12-s + (0.288 + 0.631i)13-s + (−0.000331 + 0.00230i)14-s + (−0.169 + 0.195i)15-s + (−0.146 + 0.321i)16-s + (0.388 + 0.249i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.62328 + 0.953277i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.62328 + 0.953277i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 5 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (2.35 + 4.17i)T \) |
| good | 2 | \( 1 + (-1.00 + 0.296i)T + (1.68 - 1.08i)T^{2} \) |
| 7 | \( 1 + (0.00344 - 0.00753i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-3.44 - 1.01i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.04 - 2.27i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-1.60 - 1.02i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (0.381 - 0.244i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (2.45 + 1.57i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.40 + 2.77i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.332 + 2.31i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.942 + 6.55i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.221 - 0.256i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 6.96T + 47T^{2} \) |
| 53 | \( 1 + (-4.42 + 9.68i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (4.14 + 9.08i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.17 + 1.35i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (4.10 - 1.20i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-11.2 + 3.30i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (12.4 - 7.98i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-5.87 - 12.8i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.04 - 7.23i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (2.18 + 2.51i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-1.54 - 10.7i)T + (-93.0 + 27.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.81699243496142893792961785363, −10.87946842392564680833292433028, −9.732192481220617828383132598799, −8.974224335552414638608909539930, −8.050059473165125982357927161637, −6.72776268361742984011229700174, −5.62344874823973739117803087246, −4.27476252996706089379087427360, −3.72926659527373688040118851069, −2.32087561894213823621133707175,
1.17567825956666271563177381611, 3.24012951596188030763688052997, 4.26013234569651205810834998172, 5.50788284368913415645914179312, 6.27938244517788841486237386364, 7.47376211140029311048766740209, 8.667791430532333742487687680158, 9.309478821766786903004550920762, 10.32680477980938460917658954658, 11.76300837381464495325460315938
