L(s) = 1 | + (−0.750 + 1.19i)2-s + (−0.448 − 1.67i)3-s + (−0.874 − 1.79i)4-s − 3.56i·5-s + (2.34 + 0.717i)6-s + i·7-s + (2.81 + 0.301i)8-s + (−2.59 + 1.50i)9-s + (4.27 + 2.67i)10-s + 0.335·11-s + (−2.61 + 2.26i)12-s + 3.34·13-s + (−1.19 − 0.750i)14-s + (−5.96 + 1.59i)15-s + (−2.47 + 3.14i)16-s − 0.335i·17-s + ⋯ |
L(s) = 1 | + (−0.530 + 0.847i)2-s + (−0.258 − 0.965i)3-s + (−0.437 − 0.899i)4-s − 1.59i·5-s + (0.956 + 0.292i)6-s + 0.377i·7-s + (0.994 + 0.106i)8-s + (−0.865 + 0.500i)9-s + (1.35 + 0.845i)10-s + 0.101·11-s + (−0.755 + 0.655i)12-s + 0.928·13-s + (−0.320 − 0.200i)14-s + (−1.53 + 0.412i)15-s + (−0.617 + 0.786i)16-s − 0.0813i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.655 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.609262 - 0.278152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.609262 - 0.278152i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.750 - 1.19i)T \) |
| 3 | \( 1 + (0.448 + 1.67i)T \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 3.56iT - 5T^{2} \) |
| 11 | \( 1 - 0.335T + 11T^{2} \) |
| 13 | \( 1 - 3.34T + 13T^{2} \) |
| 17 | \( 1 + 0.335iT - 17T^{2} \) |
| 19 | \( 1 + 1.84iT - 19T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 - 5.91iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 3.19T + 37T^{2} \) |
| 41 | \( 1 + 1.45iT - 41T^{2} \) |
| 43 | \( 1 + 7.49iT - 43T^{2} \) |
| 47 | \( 1 - 8.91T + 47T^{2} \) |
| 53 | \( 1 - 4.79iT - 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 + 0.353T + 61T^{2} \) |
| 67 | \( 1 - 3.19iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 4.69T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 6.89T + 83T^{2} \) |
| 89 | \( 1 - 3.87iT - 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−13.98551726114589532365316399753, −13.13403303387676268883641638676, −12.25076872741603559971806216722, −10.87569475910477147452375774802, −8.986163172981130285761828831775, −8.629199358404285528795925861595, −7.29161054844370346481905777252, −5.92686409773434031734966823308, −4.93038863201013648224285237350, −1.22071947549038401216141071232,
2.98550399227164041205278559022, 4.10712844353145207623894948623, 6.26621205603552343560941257059, 7.78276230311544649231944893930, 9.288367238494858647761631663946, 10.31578174825327489979037355894, 10.95985598067422645482632690547, 11.67400715446218811188935528182, 13.38435362051140736169804635234, 14.44560357915784377474319542498