| L(s) = 1 | + (0.623 − 0.781i)2-s + (2.57 − 0.388i)3-s + (−0.222 − 0.974i)4-s + (−0.826 + 0.563i)5-s + (1.30 − 2.25i)6-s + (1.51 + 2.61i)7-s + (−0.900 − 0.433i)8-s + (3.62 − 1.11i)9-s + (−0.0747 + 0.997i)10-s + (0.810 − 3.55i)11-s + (−0.951 − 2.42i)12-s + (0.0640 + 0.854i)13-s + (2.98 + 0.450i)14-s + (−1.90 + 1.77i)15-s + (−0.900 + 0.433i)16-s + (−2.31 − 1.58i)17-s + ⋯ |
| L(s) = 1 | + (0.440 − 0.552i)2-s + (1.48 − 0.224i)3-s + (−0.111 − 0.487i)4-s + (−0.369 + 0.251i)5-s + (0.531 − 0.921i)6-s + (0.571 + 0.989i)7-s + (−0.318 − 0.153i)8-s + (1.20 − 0.372i)9-s + (−0.0236 + 0.315i)10-s + (0.244 − 1.07i)11-s + (−0.274 − 0.700i)12-s + (0.0177 + 0.236i)13-s + (0.798 + 0.120i)14-s + (−0.493 + 0.457i)15-s + (−0.225 + 0.108i)16-s + (−0.562 − 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.700 + 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.43009 - 1.01966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43009 - 1.01966i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.623 + 0.781i)T \) |
| 5 | \( 1 + (0.826 - 0.563i)T \) |
| 43 | \( 1 + (3.87 + 5.28i)T \) |
| good | 3 | \( 1 + (-2.57 + 0.388i)T + (2.86 - 0.884i)T^{2} \) |
| 7 | \( 1 + (-1.51 - 2.61i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.810 + 3.55i)T + (-9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (-0.0640 - 0.854i)T + (-12.8 + 1.93i)T^{2} \) |
| 17 | \( 1 + (2.31 + 1.58i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-4.17 - 1.28i)T + (15.6 + 10.7i)T^{2} \) |
| 23 | \( 1 + (2.63 + 2.44i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (6.50 + 0.980i)T + (27.7 + 8.54i)T^{2} \) |
| 31 | \( 1 + (-0.225 - 0.574i)T + (-22.7 + 21.0i)T^{2} \) |
| 37 | \( 1 + (4.99 - 8.64i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.99 - 6.25i)T + (-9.12 - 39.9i)T^{2} \) |
| 47 | \( 1 + (-2.55 - 11.1i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.443 + 5.91i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (-4.99 + 2.40i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (2.21 - 5.63i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-0.544 - 0.167i)T + (55.3 + 37.7i)T^{2} \) |
| 71 | \( 1 + (-7.80 + 7.24i)T + (5.30 - 70.8i)T^{2} \) |
| 73 | \( 1 + (0.497 + 6.63i)T + (-72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (2.03 + 3.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.73 + 0.411i)T + (79.3 - 24.4i)T^{2} \) |
| 89 | \( 1 + (6.12 - 0.923i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 + (-1.51 + 6.63i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−11.39013659023637809522546807862, −10.05498624340598448523182107955, −9.028489126909434722488342592717, −8.510721629609714630659343637627, −7.62424385027629350690239230479, −6.30744835870293426634371366252, −5.05071607866004104678599021722, −3.68300317518698187164855890789, −2.90360976115241516898669752154, −1.81562701275449109662511092347,
1.97927933506483745043124442332, 3.60816713769711271763544868814, 4.15290364360968539358580304710, 5.28620626434869975986940206662, 7.12147635827510054347513936280, 7.50384792878942613609738360395, 8.406890117868289090286613749456, 9.244834792445483913331621030997, 10.13714379530335640111994663156, 11.31802660913405204336303879011
