| L(s) = 1 | + (−1.27 − 0.605i)2-s + (−0.751 + 1.61i)3-s + (1.26 + 1.54i)4-s + (0.697 − 0.488i)5-s + (1.93 − 1.60i)6-s + (1.27 + 2.21i)7-s + (−0.682 − 2.74i)8-s + (−0.104 − 0.124i)9-s + (−1.18 + 0.201i)10-s + (−0.0206 − 0.0771i)11-s + (−3.44 + 0.878i)12-s + (−0.500 − 1.07i)13-s + (−0.293 − 3.60i)14-s + (0.262 + 1.49i)15-s + (−0.789 + 3.92i)16-s + (−1.69 − 1.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.903 − 0.428i)2-s + (−0.433 + 0.930i)3-s + (0.633 + 0.773i)4-s + (0.311 − 0.218i)5-s + (0.790 − 0.655i)6-s + (0.482 + 0.836i)7-s + (−0.241 − 0.970i)8-s + (−0.0348 − 0.0415i)9-s + (−0.375 + 0.0638i)10-s + (−0.00623 − 0.0232i)11-s + (−0.994 + 0.253i)12-s + (−0.138 − 0.297i)13-s + (−0.0783 − 0.962i)14-s + (0.0679 + 0.385i)15-s + (−0.197 + 0.980i)16-s + (−0.410 − 0.344i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.578917 + 0.509542i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.578917 + 0.509542i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.27 + 0.605i)T \) |
| 19 | \( 1 + (-1.24 - 4.17i)T \) |
| good | 3 | \( 1 + (0.751 - 1.61i)T + (-1.92 - 2.29i)T^{2} \) |
| 5 | \( 1 + (-0.697 + 0.488i)T + (1.71 - 4.69i)T^{2} \) |
| 7 | \( 1 + (-1.27 - 2.21i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0206 + 0.0771i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (0.500 + 1.07i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (1.69 + 1.42i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.869 - 4.93i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.28 - 0.112i)T + (28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-2.07 - 3.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.759 - 0.759i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.07 + 0.390i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.52 + 3.60i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-5.67 - 6.76i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (8.97 + 6.28i)T + (18.1 + 49.8i)T^{2} \) |
| 59 | \( 1 + (0.797 + 9.11i)T + (-58.1 + 10.2i)T^{2} \) |
| 61 | \( 1 + (-6.20 - 4.34i)T + (20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (-0.0240 + 0.275i)T + (-65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-4.26 - 0.751i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.98 + 5.45i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (5.86 - 2.13i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.53 + 13.1i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-13.7 - 4.99i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (0.531 - 0.632i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−11.61633639150297907545088841905, −10.92641507066695384183445983415, −9.967199246470661971160545772921, −9.367059028842729033112814673893, −8.405062307923856485629689084420, −7.37570671850161280134599757446, −5.85786652008159508047058168312, −4.90072738578589518804415237660, −3.43253996808649338568800102108, −1.83051567247584987150907907344,
0.828551409223604245083800378570, 2.25432213773300400287188836231, 4.56052264916411659822959833217, 6.04960470377809881566154240948, 6.79215207499200962879284617672, 7.48756773003388279021441589496, 8.485111676366559425821594799021, 9.630142955081395988821828497975, 10.57398633738601374039332321001, 11.32645191160306198127181794990
