| L(s) = 1 | + (−0.358 − 0.933i)2-s + (1.40 + 2.15i)3-s + (−0.743 + 0.669i)4-s + (−2.15 − 0.592i)5-s + (1.51 − 2.08i)6-s + (−1.14 − 2.38i)7-s + (0.891 + 0.453i)8-s + (−1.47 + 3.31i)9-s + (0.219 + 2.22i)10-s + (2.94 − 1.52i)11-s + (−2.48 − 0.666i)12-s + (−5.30 − 0.840i)13-s + (−1.81 + 1.92i)14-s + (−1.74 − 5.48i)15-s + (0.104 − 0.994i)16-s + (2.19 − 5.71i)17-s + ⋯ |
| L(s) = 1 | + (−0.253 − 0.660i)2-s + (0.809 + 1.24i)3-s + (−0.371 + 0.334i)4-s + (−0.964 − 0.264i)5-s + (0.617 − 0.850i)6-s + (−0.434 − 0.900i)7-s + (0.315 + 0.160i)8-s + (−0.491 + 1.10i)9-s + (0.0694 + 0.703i)10-s + (0.887 − 0.461i)11-s + (−0.717 − 0.192i)12-s + (−1.47 − 0.233i)13-s + (−0.484 + 0.514i)14-s + (−0.450 − 1.41i)15-s + (0.0261 − 0.248i)16-s + (0.532 − 1.38i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.826485 - 0.738146i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.826485 - 0.738146i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.358 + 0.933i)T \) |
| 5 | \( 1 + (2.15 + 0.592i)T \) |
| 7 | \( 1 + (1.14 + 2.38i)T \) |
| 11 | \( 1 + (-2.94 + 1.52i)T \) |
| good | 3 | \( 1 + (-1.40 - 2.15i)T + (-1.22 + 2.74i)T^{2} \) |
| 13 | \( 1 + (5.30 + 0.840i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-2.19 + 5.71i)T + (-12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-3.44 + 3.82i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-0.00828 - 0.00221i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (2.01 + 0.654i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-6.30 + 0.663i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (5.74 + 3.72i)T + (15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (8.17 - 2.65i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-5.91 + 5.91i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.115 - 2.20i)T + (-46.7 + 4.91i)T^{2} \) |
| 53 | \( 1 + (-2.59 - 2.09i)T + (11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-0.731 - 0.811i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-1.83 - 0.193i)T + (59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (0.771 - 0.206i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (4.54 + 3.30i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.777 + 14.8i)T + (-72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (-2.01 + 4.53i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (1.85 + 11.7i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (4.35 + 7.54i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.972 - 6.14i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−9.973962656265485668652115209722, −9.395607802126033656704996332545, −8.790592142076516336050507343765, −7.66008540550382967366249046866, −7.07567747811557372123352631757, −5.03342843119184666580898902334, −4.40736080916100568786509625720, −3.46397521849721335981275088497, −2.89577697799150796904317374420, −0.58188637033070856648401221082,
1.54899994218771839920604400684, 2.83583970461118582453947867450, 3.97551132780801203289725469158, 5.40952645317547907274471642713, 6.64847720917918426269236879230, 7.02793023387438169994401182592, 8.029074316255536915995088416336, 8.378018277337227826330432152345, 9.425008657121267668590489755845, 10.18812570949871298980853988162
