| L(s) = 1 | + (−0.5 − 0.363i)2-s + (0.309 + 0.951i)3-s + (−0.5 − 1.53i)4-s + (−1.80 + 1.31i)5-s + (0.190 − 0.587i)6-s − 1.61·7-s + (−0.690 + 2.12i)8-s + (1.61 − 1.17i)9-s + 1.38·10-s + (0.618 + 0.449i)11-s + (1.30 − 0.951i)12-s + (3.92 − 2.85i)13-s + (0.809 + 0.587i)14-s + (−1.80 − 1.31i)15-s + (−1.49 + 1.08i)16-s + (−0.236 + 0.726i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.256i)2-s + (0.178 + 0.549i)3-s + (−0.250 − 0.769i)4-s + (−0.809 + 0.587i)5-s + (0.0779 − 0.239i)6-s − 0.611·7-s + (−0.244 + 0.751i)8-s + (0.539 − 0.391i)9-s + 0.437·10-s + (0.186 + 0.135i)11-s + (0.377 − 0.274i)12-s + (1.08 − 0.791i)13-s + (0.216 + 0.157i)14-s + (−0.467 − 0.339i)15-s + (−0.374 + 0.272i)16-s + (−0.0572 + 0.176i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.529534 - 0.0333154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.529534 - 0.0333154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.80 - 1.31i)T \) |
| good | 2 | \( 1 + (0.5 + 0.363i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.309 - 0.951i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + (-0.618 - 0.449i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.92 + 2.85i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.236 - 0.726i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.80 - 5.56i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (6.66 + 4.84i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.427 + 1.31i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.927 - 2.85i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (3.42 - 2.48i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.23 + 3.07i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 1.85T + 43T^{2} \) |
| 47 | \( 1 + (0.5 + 1.53i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.69 - 5.20i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.35 + 2.43i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.80 - 2.76i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.85 + 8.78i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.35 + 4.16i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.28 - 5.29i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.954 - 2.93i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.545 - 1.67i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (7.23 + 5.25i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.881 - 2.71i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−18.09369935029835365181006296283, −16.11870739004231735436644807772, −15.27296137164702920299758227524, −14.23329304054160060706842680390, −12.41568399753153466766064232053, −10.73364131395871901679656565556, −9.970660453849164957529609063316, −8.400323665832239832022596326221, −6.27578904102493561363949704096, −3.87632635392220914344699637256,
4.01305883699761872997695967334, 6.88842797735002891028014004884, 8.117834134407304095731906220012, 9.275463700894892957584831283175, 11.51714826958993390227905514135, 12.78745959025090485470809083746, 13.57080781772427432924421874687, 15.83299224852732678802708324411, 16.28658582747629304639915398031, 17.73694771594256913984512259039
