L(s) = 1 | + (0.707 + 0.707i)2-s + (1.31 + 1.12i)3-s + 1.00i·4-s + (0.192 + 2.22i)5-s + (0.133 + 1.72i)6-s + (1.96 − 1.96i)7-s + (−0.707 + 0.707i)8-s + (0.461 + 2.96i)9-s + (−1.43 + 1.71i)10-s + 2.97i·11-s + (−1.12 + 1.31i)12-s + (−1.33 − 1.33i)13-s + 2.77·14-s + (−2.25 + 3.14i)15-s − 1.00·16-s + (−0.217 − 0.217i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.759 + 0.650i)3-s + 0.500i·4-s + (0.0860 + 0.996i)5-s + (0.0545 + 0.704i)6-s + (0.740 − 0.740i)7-s + (−0.250 + 0.250i)8-s + (0.153 + 0.988i)9-s + (−0.455 + 0.541i)10-s + 0.896i·11-s + (−0.325 + 0.379i)12-s + (−0.369 − 0.369i)13-s + 0.740·14-s + (−0.582 + 0.812i)15-s − 0.250·16-s + (−0.0527 − 0.0527i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45876 + 2.18917i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45876 + 2.18917i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 3 | \( 1 + (-1.31 - 1.12i)T \) |
| 5 | \( 1 + (-0.192 - 2.22i)T \) |
| 23 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-1.96 + 1.96i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.97iT - 11T^{2} \) |
| 13 | \( 1 + (1.33 + 1.33i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.217 + 0.217i)T + 17iT^{2} \) |
| 19 | \( 1 + 6.42iT - 19T^{2} \) |
| 29 | \( 1 - 1.22T + 29T^{2} \) |
| 31 | \( 1 - 6.33T + 31T^{2} \) |
| 37 | \( 1 + (-1.43 + 1.43i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.04iT - 41T^{2} \) |
| 43 | \( 1 + (1.80 + 1.80i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.199 + 0.199i)T + 47iT^{2} \) |
| 53 | \( 1 + (9.17 - 9.17i)T - 53iT^{2} \) |
| 59 | \( 1 + 9.43T + 59T^{2} \) |
| 61 | \( 1 - 5.57T + 61T^{2} \) |
| 67 | \( 1 + (-9.41 + 9.41i)T - 67iT^{2} \) |
| 71 | \( 1 + 7.64iT - 71T^{2} \) |
| 73 | \( 1 + (4.05 + 4.05i)T + 73iT^{2} \) |
| 79 | \( 1 - 14.4iT - 79T^{2} \) |
| 83 | \( 1 + (-12.1 + 12.1i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.45T + 89T^{2} \) |
| 97 | \( 1 + (-8.98 + 8.98i)T - 97iT^{2} \) |
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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
−10.68999408490292294339805504615, −9.926996255226923173863799861170, −9.002203228350635285156994868260, −7.80915305323603333831965999031, −7.40076265460166880115329556866, −6.45623643586689759414818005085, −4.94742628696886560977567049944, −4.43283820690198843636094940727, −3.23660465449134863536833105584, −2.27511675648009942592636233567,
1.23320605176857921543766175982, 2.21145340140973450052379762515, 3.46160179258334671895735838827, 4.60463265729437382417147089265, 5.60124275937955345485966574948, 6.45423236866594418622810857683, 7.999269845338308302222312644920, 8.360659213170584389513336222576, 9.261798704223673164082568657829, 10.06352400580875969962231591126