L(s) = 1 | + (−0.870 + 0.492i)2-s + (0.899 + 0.843i)3-s + (0.515 − 0.856i)4-s + (−1.83 − 0.0675i)5-s + (−1.19 − 0.291i)6-s + (−0.146 − 0.751i)7-s + (−0.0275 + 0.999i)8-s + (−0.0951 − 1.47i)9-s + (1.63 − 0.844i)10-s + (1.26 + 1.72i)11-s + (1.18 − 0.335i)12-s + (−0.973 − 0.416i)13-s + (0.497 + 0.581i)14-s + (−1.59 − 1.60i)15-s + (−0.467 − 0.883i)16-s + (6.56 − 0.120i)17-s + ⋯ |
L(s) = 1 | + (−0.615 + 0.347i)2-s + (0.519 + 0.486i)3-s + (0.257 − 0.428i)4-s + (−0.821 − 0.0301i)5-s + (−0.488 − 0.119i)6-s + (−0.0554 − 0.283i)7-s + (−0.00974 + 0.353i)8-s + (−0.0317 − 0.492i)9-s + (0.516 − 0.267i)10-s + (0.381 + 0.518i)11-s + (0.342 − 0.0968i)12-s + (−0.269 − 0.115i)13-s + (0.132 + 0.155i)14-s + (−0.411 − 0.415i)15-s + (−0.116 − 0.220i)16-s + (1.59 − 0.0292i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11346 + 0.469524i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11346 + 0.469524i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.870 - 0.492i)T \) |
| 19 | \( 1 + (-1.55 - 4.07i)T \) |
good | 3 | \( 1 + (-0.899 - 0.843i)T + (0.192 + 2.99i)T^{2} \) |
| 5 | \( 1 + (1.83 + 0.0675i)T + (4.98 + 0.367i)T^{2} \) |
| 7 | \( 1 + (0.146 + 0.751i)T + (-6.48 + 2.63i)T^{2} \) |
| 11 | \( 1 + (-1.26 - 1.72i)T + (-3.28 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.973 + 0.416i)T + (8.97 + 9.40i)T^{2} \) |
| 17 | \( 1 + (-6.56 + 0.120i)T + (16.9 - 0.624i)T^{2} \) |
| 23 | \( 1 + (-5.49 - 1.66i)T + (19.1 + 12.7i)T^{2} \) |
| 29 | \( 1 + (-4.14 - 3.35i)T + (6.08 + 28.3i)T^{2} \) |
| 31 | \( 1 + (-0.538 - 1.43i)T + (-23.3 + 20.3i)T^{2} \) |
| 37 | \( 1 + (-0.827 + 1.88i)T + (-25.0 - 27.2i)T^{2} \) |
| 41 | \( 1 + (-5.53 - 0.510i)T + (40.3 + 7.49i)T^{2} \) |
| 43 | \( 1 + (-0.529 - 11.5i)T + (-42.8 + 3.94i)T^{2} \) |
| 47 | \( 1 + (-1.24 + 1.30i)T + (-2.15 - 46.9i)T^{2} \) |
| 53 | \( 1 + (3.66 + 12.5i)T + (-44.6 + 28.5i)T^{2} \) |
| 59 | \( 1 + (2.46 + 5.34i)T + (-38.3 + 44.8i)T^{2} \) |
| 61 | \( 1 + (5.90 - 1.20i)T + (56.0 - 23.9i)T^{2} \) |
| 67 | \( 1 + (6.20 - 0.919i)T + (64.1 - 19.4i)T^{2} \) |
| 71 | \( 1 + (-9.31 - 1.90i)T + (65.2 + 27.9i)T^{2} \) |
| 73 | \( 1 + (4.85 - 8.77i)T + (-38.7 - 61.8i)T^{2} \) |
| 79 | \( 1 + (-1.11 + 0.712i)T + (33.0 - 71.7i)T^{2} \) |
| 83 | \( 1 + (1.01 - 7.30i)T + (-79.8 - 22.5i)T^{2} \) |
| 89 | \( 1 + (2.42 + 4.38i)T + (-47.3 + 75.3i)T^{2} \) |
| 97 | \( 1 + (-11.9 - 1.76i)T + (92.8 + 28.1i)T^{2} \) |
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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
−10.10303451588504043214676495492, −9.714632134524944067663463715363, −8.803869320194866196123035077246, −7.900930431666908960695814204133, −7.34392157972756172264843334664, −6.26929326356637683261187262953, −5.05197134852231479094862476513, −3.89334030573684996869445161831, −3.07865936185027965832878811942, −1.11485811609345832304185070789,
0.971372676203778072888162084401, 2.54235174879194066550813929750, 3.36458373980351590203770164204, 4.67985526291963997899643248044, 5.98829123993302249386225784616, 7.36203432769348359408805686070, 7.62837699573870115127274444928, 8.626158469032956088654742663808, 9.214354287874849460173640024586, 10.34786559758361129520234681997