L(s) = 1 | + (1.41 + i)3-s + 2.23·5-s + (1.41 − 2.23i)7-s + (1.00 + 2.82i)9-s + 2.82i·11-s − 2.82·13-s + (3.16 + 2.23i)15-s − 4i·17-s − 6.32i·19-s + (4.23 − 1.74i)21-s − 6.32·23-s + 5.00·25-s + (−1.41 + 5.00i)27-s + 5.65i·29-s + 6.32i·31-s + ⋯ |
L(s) = 1 | + (0.816 + 0.577i)3-s + 0.999·5-s + (0.534 − 0.845i)7-s + (0.333 + 0.942i)9-s + 0.852i·11-s − 0.784·13-s + (0.816 + 0.577i)15-s − 0.970i·17-s − 1.45i·19-s + (0.924 − 0.381i)21-s − 1.31·23-s + 1.00·25-s + (−0.272 + 0.962i)27-s + 1.05i·29-s + 1.13i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.06638 + 0.409605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.06638 + 0.409605i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.41 - i)T \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 + (-1.41 + 2.23i)T \) |
good | 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 6.32iT - 19T^{2} \) |
| 23 | \( 1 + 6.32T + 23T^{2} \) |
| 29 | \( 1 - 5.65iT - 29T^{2} \) |
| 31 | \( 1 - 6.32iT - 31T^{2} \) |
| 37 | \( 1 - 8.94iT - 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 + 4.47iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 4.47iT - 67T^{2} \) |
| 71 | \( 1 + 14.1iT - 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 - 8.48T + 97T^{2} \) |
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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
−10.93458309502706860342116512601, −10.15274399969793686964040115821, −9.575260525532798933493617642047, −8.706245552884657717260095058942, −7.50885902575966121593042451398, −6.82809211382394743180246987167, −5.05190190999572118745081153950, −4.59928637444759179426049615563, −2.99823513516520413078270039891, −1.84364732706582227047180785177,
1.72942846543979059626952248260, 2.59099709857881288137073607643, 4.07392455485634343164929731370, 5.81590421982548481566368161224, 6.12023833237011833104491499726, 7.77039413393366960013992507136, 8.273694808898005189307385726450, 9.334397943187078791860312319059, 9.955122870951026136193057914608, 11.18141550147851782661106947465