| L(s) = 1 | + (0.173 − 0.984i)2-s + (2.14 + 1.79i)3-s + (−0.939 − 0.342i)4-s + (2.20 − 0.801i)5-s + (2.14 − 1.79i)6-s + (0.642 + 1.11i)7-s + (−0.5 + 0.866i)8-s + (0.834 + 4.73i)9-s + (−0.407 − 2.30i)10-s + (−2.87 + 4.98i)11-s + (−1.39 − 2.41i)12-s + (0.233 − 0.195i)13-s + (1.20 − 0.439i)14-s + (6.15 + 2.24i)15-s + (0.766 + 0.642i)16-s + (0.726 − 4.12i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (1.23 + 1.03i)3-s + (−0.469 − 0.171i)4-s + (0.985 − 0.358i)5-s + (0.873 − 0.733i)6-s + (0.242 + 0.420i)7-s + (−0.176 + 0.306i)8-s + (0.278 + 1.57i)9-s + (−0.128 − 0.730i)10-s + (−0.867 + 1.50i)11-s + (−0.403 − 0.698i)12-s + (0.0646 − 0.0542i)13-s + (0.322 − 0.117i)14-s + (1.58 + 0.578i)15-s + (0.191 + 0.160i)16-s + (0.176 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.61789 + 0.371607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.61789 + 0.371607i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-2.14 - 1.79i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-2.20 + 0.801i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.642 - 1.11i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.87 - 4.98i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.233 + 0.195i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.726 + 4.12i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-6.08 - 2.21i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.543 + 3.08i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.22 + 5.57i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.97T + 37T^{2} \) |
| 41 | \( 1 + (3.84 + 3.22i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.929 + 0.338i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.763 + 4.32i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (3.09 + 1.12i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.575 + 3.26i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 3.74i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.760 + 4.31i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-4.14 + 1.51i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (1.73 + 1.45i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-6.57 - 5.51i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.88 - 8.45i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (11.5 - 9.69i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-3.01 + 17.0i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.07497983266626205167421262721, −9.632057896607184498957168587338, −9.148624789204534549161760049529, −8.215198192651976025442171148580, −7.18354399177779020862798416008, −5.32470511995315233564445236541, −4.98765339994941273270235106348, −3.81024181556985283889328422171, −2.61017227556318579301906236802, −1.98715231312459898510281601223,
1.34534573076917155006060382409, 2.70642423451989605129061796851, 3.52918270059359469595766354846, 5.21921346251970932836117358326, 6.17804411057150903958615721248, 6.91303048060660488013002228982, 7.80609863867173119975758893234, 8.510302606026462434163623143503, 9.056996971755859706508639765120, 10.29821618879043171794555079696
