| 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
−11.31802660913405204336303879011, −10.13714379530335640111994663156, −9.244834792445483913331621030997, −8.406890117868289090286613749456, −7.50384792878942613609738360395, −7.12147635827510054347513936280, −5.28620626434869975986940206662, −4.15290364360968539358580304710, −3.60816713769711271763544868814, −1.97927933506483745043124442332,
1.81562701275449109662511092347, 2.90360976115241516898669752154, 3.68300317518698187164855890789, 5.05071607866004104678599021722, 6.30744835870293426634371366252, 7.62424385027629350690239230479, 8.510721629609714630659343637627, 9.028489126909434722488342592717, 10.05498624340598448523182107955, 11.39013659023637809522546807862
