| 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
−10.18812570949871298980853988162, −9.425008657121267668590489755845, −8.378018277337227826330432152345, −8.029074316255536915995088416336, −7.02793023387438169994401182592, −6.64847720917918426269236879230, −5.40952645317547907274471642713, −3.97551132780801203289725469158, −2.83583970461118582453947867450, −1.54899994218771839920604400684,
0.58188637033070856648401221082, 2.89577697799150796904317374420, 3.46397521849721335981275088497, 4.40736080916100568786509625720, 5.03342843119184666580898902334, 7.07567747811557372123352631757, 7.66008540550382967366249046866, 8.790592142076516336050507343765, 9.395607802126033656704996332545, 9.973962656265485668652115209722
