| L(s) = 1 | + (−0.654 − 0.755i)2-s + (0.841 + 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (−0.142 − 0.989i)6-s + (2.52 + 0.740i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (3.03 − 3.49i)11-s + (−0.654 + 0.755i)12-s + (−2.32 + 0.682i)13-s + (−1.09 − 2.39i)14-s + (0.841 − 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.246 + 1.71i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (0.485 + 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.185 − 0.406i)5-s + (−0.0580 − 0.404i)6-s + (0.953 + 0.280i)7-s + (0.297 − 0.191i)8-s + (0.138 + 0.303i)9-s + (−0.303 + 0.0890i)10-s + (0.914 − 1.05i)11-s + (−0.189 + 0.218i)12-s + (−0.644 + 0.189i)13-s + (−0.291 − 0.639i)14-s + (0.217 − 0.139i)15-s + (−0.239 − 0.0704i)16-s + (0.0597 + 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59261 - 0.420513i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59261 - 0.420513i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 3 | \( 1 + (-0.841 - 0.540i)T \) |
| 5 | \( 1 + (-0.415 + 0.909i)T \) |
| 23 | \( 1 + (-3.48 + 3.29i)T \) |
| good | 7 | \( 1 + (-2.52 - 0.740i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-3.03 + 3.49i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.32 - 0.682i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.246 - 1.71i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.221 - 1.53i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.675 - 4.69i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-0.508 + 0.326i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (0.955 + 2.09i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.21 + 7.03i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.14 - 3.30i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 53 | \( 1 + (-7.92 - 2.32i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (3.27 - 0.962i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.54 + 0.991i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (0.0237 + 0.0273i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (10.3 + 11.8i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.31 - 9.16i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (4.05 - 1.18i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.05 - 6.68i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-6.01 - 3.86i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.932 + 2.04i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.47302107334737948609765061970, −9.320423177591768477934827207311, −8.805765608964490730081596064337, −8.184687026056960937603455516571, −7.12793596957866581547326855883, −5.79336073662879471393316871672, −4.70408480035732687344993179806, −3.74514811031517635065334821215, −2.47499075549950928948468363326, −1.25035223280734391839987606209,
1.35567438185904620379046281052, 2.55448892691533672072606011557, 4.18626342526042937555479174437, 5.11190468959820347681202030960, 6.39463453995646374866142962279, 7.30710823226515797663957397989, 7.66773879298967071117392156373, 8.827805370894520293079351496752, 9.542407461282852481825103597575, 10.29337681006221219439879679820
