| L(s) = 1 | + (−1.33 + 0.458i)2-s + (1.57 − 1.22i)4-s + 2.45i·5-s + (−1.55 + 2.36i)8-s + (−1.12 − 3.28i)10-s − 1.26i·11-s − 2.99i·13-s + (0.992 − 3.87i)16-s − 1.83i·17-s − 4.15·19-s + (3.00 + 3.87i)20-s + (0.579 + 1.69i)22-s + 6.73i·23-s − 1.01·25-s + (1.37 + 4.01i)26-s + ⋯ |
| L(s) = 1 | + (−0.946 + 0.324i)2-s + (0.789 − 0.613i)4-s + 1.09i·5-s + (−0.548 + 0.836i)8-s + (−0.355 − 1.03i)10-s − 0.381i·11-s − 0.831i·13-s + (0.248 − 0.968i)16-s − 0.444i·17-s − 0.954·19-s + (0.672 + 0.866i)20-s + (0.123 + 0.360i)22-s + 1.40i·23-s − 0.203·25-s + (0.269 + 0.786i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.026783317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.026783317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 - 0.458i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.45iT - 5T^{2} \) |
| 11 | \( 1 + 1.26iT - 11T^{2} \) |
| 13 | \( 1 + 2.99iT - 13T^{2} \) |
| 17 | \( 1 + 1.83iT - 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 - 6.73iT - 23T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 - 9.43T + 31T^{2} \) |
| 37 | \( 1 - 7.51T + 37T^{2} \) |
| 41 | \( 1 + 1.08iT - 41T^{2} \) |
| 43 | \( 1 - 6.27iT - 43T^{2} \) |
| 47 | \( 1 + 7.35T + 47T^{2} \) |
| 53 | \( 1 - 0.0716T + 53T^{2} \) |
| 59 | \( 1 + 3.36T + 59T^{2} \) |
| 61 | \( 1 - 11.1iT - 61T^{2} \) |
| 67 | \( 1 - 2.80iT - 67T^{2} \) |
| 71 | \( 1 + 2.92iT - 71T^{2} \) |
| 73 | \( 1 - 8.10iT - 73T^{2} \) |
| 79 | \( 1 + 1.78iT - 79T^{2} \) |
| 83 | \( 1 + 5.33T + 83T^{2} \) |
| 89 | \( 1 - 8.57iT - 89T^{2} \) |
| 97 | \( 1 - 7.10iT - 97T^{2} \) |
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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
−9.617735132556913543689119208584, −8.485756266182139228570330160473, −7.998119735495639890335409386196, −7.12908903960980877438336352594, −6.42679044276323688224900135928, −5.82313990211807059710024320541, −4.65106525959088450759166986580, −3.11901533270770204872409234978, −2.58335826267009714571580592636, −1.02678828585120226327845881530,
0.65064553807999743435787494818, 1.79288621517193460399551757546, 2.79841788073755457994066577283, 4.26588102552327622189739657257, 4.73013222036656433203961844039, 6.31443761899482464277952819267, 6.65692027768862067903024281659, 7.956850683481610279096563183058, 8.467694648957924352183284458669, 8.986750690300143872602832454299
