L(s) = 1 | + (19.4 + 25.3i)2-s + 120. i·3-s + (−265. + 988. i)4-s + 2.48e3·5-s + (−3.05e3 + 2.34e3i)6-s − 2.91e4i·7-s + (−3.02e4 + 1.25e4i)8-s + 4.45e4·9-s + (4.84e4 + 6.31e4i)10-s + 8.84e4i·11-s + (−1.18e5 − 3.19e4i)12-s − 3.00e5·13-s + (7.39e5 − 5.67e5i)14-s + 2.99e5i·15-s + (−9.07e5 − 5.25e5i)16-s − 1.52e5·17-s + ⋯ |
L(s) = 1 | + (0.608 + 0.793i)2-s + 0.494i·3-s + (−0.259 + 0.965i)4-s + 0.795·5-s + (−0.392 + 0.301i)6-s − 1.73i·7-s + (−0.924 + 0.381i)8-s + 0.755·9-s + (0.484 + 0.631i)10-s + 0.549i·11-s + (−0.477 − 0.128i)12-s − 0.809·13-s + (1.37 − 1.05i)14-s + 0.393i·15-s + (−0.865 − 0.501i)16-s − 0.107·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.44565 + 1.10855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44565 + 1.10855i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-19.4 - 25.3i)T \) |
good | 3 | \( 1 - 120. iT - 5.90e4T^{2} \) |
| 5 | \( 1 - 2.48e3T + 9.76e6T^{2} \) |
| 7 | \( 1 + 2.91e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 8.84e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 3.00e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 1.52e5T + 2.01e12T^{2} \) |
| 19 | \( 1 + 4.76e4iT - 6.13e12T^{2} \) |
| 23 | \( 1 + 7.60e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 6.49e6T + 4.20e14T^{2} \) |
| 31 | \( 1 - 3.06e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 5.72e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 2.49e7T + 1.34e16T^{2} \) |
| 43 | \( 1 - 1.22e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 - 1.03e7iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 5.65e8T + 1.74e17T^{2} \) |
| 59 | \( 1 + 3.70e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 3.89e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 1.36e9iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 1.93e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 3.26e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 2.38e8iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 6.66e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 2.76e9T + 3.11e19T^{2} \) |
| 97 | \( 1 - 1.47e9T + 7.37e19T^{2} \) |
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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
−23.22664503655558199168076198860, −21.79510207899135475795331149790, −20.47028879302976326012815410521, −17.59707180947016289892955724082, −16.42444895783949711496739066423, −14.51262165287269867087373644324, −13.08228593701896098298789512989, −10.05123483445908925802978314686, −7.10839456140413286653479066916, −4.43798841756105197179041122545,
2.14674351421075907605324664010, 5.72392889532439134048585383642, 9.529020991482436045869075689001, 11.94507792642790259188233467505, 13.33794713020308486638585085897, 15.21488031710711449531718585184, 18.12988880105713957401230650970, 19.19146084245894350801067916891, 21.36507386070773267698383403046, 22.03249571383238934986586655325