| L(s) = 1 | + (0.222 + 0.974i)2-s + (−1.02 − 0.947i)3-s + (−0.900 + 0.433i)4-s + (−0.988 + 0.149i)5-s + (0.696 − 1.20i)6-s + (2.13 + 3.69i)7-s + (−0.623 − 0.781i)8-s + (−0.0790 − 1.05i)9-s + (−0.365 − 0.930i)10-s + (−3.17 − 1.52i)11-s + (1.33 + 0.410i)12-s + (−1.57 + 4.00i)13-s + (−3.12 + 2.89i)14-s + (1.15 + 0.785i)15-s + (0.623 − 0.781i)16-s + (−2.00 − 0.302i)17-s + ⋯ |
| L(s) = 1 | + (0.157 + 0.689i)2-s + (−0.589 − 0.547i)3-s + (−0.450 + 0.216i)4-s + (−0.442 + 0.0666i)5-s + (0.284 − 0.492i)6-s + (0.805 + 1.39i)7-s + (−0.220 − 0.276i)8-s + (−0.0263 − 0.351i)9-s + (−0.115 − 0.294i)10-s + (−0.956 − 0.460i)11-s + (0.384 + 0.118i)12-s + (−0.435 + 1.11i)13-s + (−0.834 + 0.774i)14-s + (0.297 + 0.202i)15-s + (0.155 − 0.195i)16-s + (−0.486 − 0.0732i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.148417 + 0.627235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.148417 + 0.627235i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (0.988 - 0.149i)T \) |
| 43 | \( 1 + (-6.55 + 0.184i)T \) |
| good | 3 | \( 1 + (1.02 + 0.947i)T + (0.224 + 2.99i)T^{2} \) |
| 7 | \( 1 + (-2.13 - 3.69i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.17 + 1.52i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (1.57 - 4.00i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (2.00 + 0.302i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (0.541 - 7.23i)T + (-18.7 - 2.83i)T^{2} \) |
| 23 | \( 1 + (-0.139 + 0.0951i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (7.42 - 6.88i)T + (2.16 - 28.9i)T^{2} \) |
| 31 | \( 1 + (6.29 + 1.94i)T + (25.6 + 17.4i)T^{2} \) |
| 37 | \( 1 + (-0.391 + 0.678i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.04 - 4.59i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (-10.2 + 4.91i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (4.91 + 12.5i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (-4.29 + 5.38i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-3.15 + 0.971i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (0.443 - 5.91i)T + (-66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (-13.4 - 9.15i)T + (25.9 + 66.0i)T^{2} \) |
| 73 | \( 1 + (1.33 - 3.39i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (6.25 + 10.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.7 - 11.8i)T + (6.20 + 82.7i)T^{2} \) |
| 89 | \( 1 + (6.95 + 6.45i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + (1.49 + 0.719i)T + (60.4 + 75.8i)T^{2} \) |
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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
−11.64754918727537459825570277382, −11.00297115706795037235749661660, −9.436790501689041580633927804994, −8.635263972402337516117971290822, −7.77860942396316702902790039256, −6.86203685423139873657927613096, −5.74984423068741011361488165530, −5.26224570204374365690158115523, −3.79699690178486777795106597825, −2.03864149966129967294863051107,
0.40533747058377246747357597623, 2.43475245478083032704773478767, 4.05941136992874009867708739400, 4.73142507132852949721206568088, 5.52236854073856926417754762551, 7.39956449135361601987765249794, 7.79784835810932464131754786995, 9.237482069307579826548852015684, 10.35478293242689797322086584762, 10.90049939759401024893076697264
