L(s) = 1 | + (0.809 + 0.587i)2-s + (0.913 + 0.406i)3-s + (0.309 + 0.951i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−1.88 + 2.09i)7-s + (−0.309 + 0.951i)8-s + (0.669 + 0.743i)9-s + (−0.913 + 0.406i)10-s + (−1.23 − 0.263i)11-s + (−0.104 + 0.994i)12-s + (0.0140 + 0.134i)13-s + (−2.75 + 0.586i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (3.94 − 0.838i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (0.527 + 0.234i)3-s + (0.154 + 0.475i)4-s + (−0.223 + 0.387i)5-s + (0.204 + 0.353i)6-s + (−0.713 + 0.792i)7-s + (−0.109 + 0.336i)8-s + (0.223 + 0.247i)9-s + (−0.288 + 0.128i)10-s + (−0.373 − 0.0793i)11-s + (−0.0301 + 0.287i)12-s + (0.00391 + 0.0372i)13-s + (−0.737 + 0.156i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (0.956 − 0.203i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.729355 + 1.84576i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.729355 + 1.84576i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.913 - 0.406i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (3.12 - 4.60i)T \) |
good | 7 | \( 1 + (1.88 - 2.09i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (1.23 + 0.263i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.0140 - 0.134i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-3.94 + 0.838i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.316 - 3.01i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (2.44 - 7.51i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (4.11 + 2.99i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (2.33 + 4.03i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.621 + 0.276i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (-0.209 + 1.99i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-6.79 + 4.93i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.59 - 2.87i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-11.2 - 5.01i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 0.424T + 61T^{2} \) |
| 67 | \( 1 + (-3.23 + 5.60i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.64 - 4.05i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-8.63 - 1.83i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (4.62 - 0.982i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-3.59 + 1.60i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-2.89 - 8.90i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.59 + 4.91i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.23112138776861889352627332641, −9.519824301489662166974219752146, −8.647059507522385090351004516184, −7.72067109678909167001916932075, −7.09823240064029881090849702509, −5.83536325493059854853313175276, −5.40323030909600406424805691861, −3.86821036100342673908972212502, −3.30278698832142581185865189709, −2.18240810359331951146458374570,
0.70997346990553416014128117349, 2.27317585200507516249295615883, 3.40087285932341295294180173793, 4.14723607972679137059730801564, 5.20591279846380747692084824395, 6.32697038986914393967195423194, 7.18233549795230985929323182104, 8.020696870517917923510241052697, 8.997446016030172825042615108921, 9.903543908317660050448128534773