| L(s) = 1 | + (−1.98 + 0.644i)2-s + (1.29 + 1.77i)3-s + (1.89 − 1.38i)4-s + (−1.22 − 1.87i)5-s + (−3.70 − 2.69i)6-s − 0.992i·7-s + (−0.427 + 0.587i)8-s + (−0.566 + 1.74i)9-s + (3.63 + 2.91i)10-s + (0.618 + 1.90i)11-s + (4.91 + 1.59i)12-s + (−3.20 − 1.04i)13-s + (0.639 + 1.96i)14-s + (1.74 − 4.59i)15-s + (−0.983 + 3.02i)16-s + (−1.70 + 2.34i)17-s + ⋯ |
| L(s) = 1 | + (−1.40 + 0.455i)2-s + (0.746 + 1.02i)3-s + (0.949 − 0.690i)4-s + (−0.548 − 0.836i)5-s + (−1.51 − 1.10i)6-s − 0.375i·7-s + (−0.150 + 0.207i)8-s + (−0.188 + 0.581i)9-s + (1.14 + 0.923i)10-s + (0.186 + 0.573i)11-s + (1.41 + 0.460i)12-s + (−0.889 − 0.289i)13-s + (0.170 + 0.526i)14-s + (0.449 − 1.18i)15-s + (−0.245 + 0.756i)16-s + (−0.412 + 0.567i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.614 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.378458 + 0.184915i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378458 + 0.184915i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.22 + 1.87i)T \) |
| good | 2 | \( 1 + (1.98 - 0.644i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-1.29 - 1.77i)T + (-0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 0.992iT - 7T^{2} \) |
| 11 | \( 1 + (-0.618 - 1.90i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (3.20 + 1.04i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.70 - 2.34i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.09 + 1.51i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-4.32 + 1.40i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (4.35 - 3.16i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.110 + 0.0802i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.04 - 0.664i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.66 + 8.21i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 4.64iT - 43T^{2} \) |
| 47 | \( 1 + (-5.83 - 8.03i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (4.44 + 6.12i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.51 - 4.67i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.855 + 2.63i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (1.28 - 1.76i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (7.80 - 5.66i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.737 - 0.239i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-12.8 + 9.31i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.04 - 1.43i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.48 + 13.7i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (10.0 + 13.7i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−17.50913434106713382839824230140, −16.71509611894711969631319882391, −15.62447601374591286540020049584, −14.83373849092451279656091434944, −12.76836824997649359404149923077, −10.70478661265070728555648783389, −9.484048326427187376772211246516, −8.700756789123480086234663064578, −7.37214446289042285892258381094, −4.36048904079619467511194010037,
2.51905553356217036605454282958, 7.04203274766215539700169588614, 8.022729140610343966777265765300, 9.264606005725786684581250817764, 10.89256812402103518257817939439, 12.05295312665352228041124940876, 13.75654246736096946528183602559, 15.00438710068842520593243068550, 16.71570520657099980445482553559, 18.06521330098199008199030446471
