| L(s) = 1 | + (0.173 − 0.984i)2-s + (0.983 + 0.825i)3-s + (−0.939 − 0.342i)4-s + (−3.47 + 1.26i)5-s + (0.983 − 0.825i)6-s + (0.221 + 0.383i)7-s + (−0.5 + 0.866i)8-s + (−0.234 − 1.33i)9-s + (0.641 + 3.63i)10-s + (2.01 − 3.48i)11-s + (−0.642 − 1.11i)12-s + (3.74 − 3.14i)13-s + (0.415 − 0.151i)14-s + (−4.45 − 1.62i)15-s + (0.766 + 0.642i)16-s + (0.0463 − 0.262i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (0.567 + 0.476i)3-s + (−0.469 − 0.171i)4-s + (−1.55 + 0.565i)5-s + (0.401 − 0.336i)6-s + (0.0836 + 0.144i)7-s + (−0.176 + 0.306i)8-s + (−0.0782 − 0.443i)9-s + (0.202 + 1.15i)10-s + (0.607 − 1.05i)11-s + (−0.185 − 0.321i)12-s + (1.03 − 0.872i)13-s + (0.111 − 0.0404i)14-s + (−1.15 − 0.419i)15-s + (0.191 + 0.160i)16-s + (0.0112 − 0.0637i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.14029 - 0.840247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14029 - 0.840247i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.983 - 0.825i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (3.47 - 1.26i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.221 - 0.383i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.01 + 3.48i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.74 + 3.14i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0463 + 0.262i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-8.65 - 3.14i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.0388 + 0.220i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (1.73 + 3.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.44T + 37T^{2} \) |
| 41 | \( 1 + (6.02 + 5.05i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-5.29 + 1.92i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.381 - 2.16i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (9.34 + 3.40i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.609 - 3.45i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (3.83 + 1.39i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.0256 - 0.145i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (10.7 - 3.92i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-1.08 - 0.914i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.81 - 6.55i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.64 - 2.84i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.56 - 3.83i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.709 + 4.02i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.50734448355061122619900769332, −9.235751389495894138883363748049, −8.654012088418148725944905650841, −7.931118617318502376443008263545, −6.79340081044646493244699926366, −5.61172061000398819630385479833, −4.20953777667054336420671906851, −3.42775984082986075024790837140, −3.08843082738602545776778225967, −0.78566396627995879363134002800,
1.37462336301577599028008828585, 3.23563430699650815713235180306, 4.33806686603672842515163373213, 4.86005138865167089498872989748, 6.51585963609099087202538486013, 7.27052134231216334578658797135, 7.85632012478255865150821271192, 8.752869332453460004308817732845, 9.135131133376895829952180252649, 10.74431376240575812281210555250
