L(s) = 1 | + (−0.763 + 0.763i)5-s + 1.33·7-s + (−1.95 − 1.95i)11-s + (−4.18 + 4.18i)13-s − 4.03i·17-s + (4.26 + 4.26i)19-s + 8.86i·23-s + 3.83i·25-s + (−1.23 − 1.23i)29-s + 2.87i·31-s + (−1.02 + 1.02i)35-s + (−0.434 − 0.434i)37-s − 7.81·41-s + (5.49 − 5.49i)43-s − 3.20·47-s + ⋯ |
L(s) = 1 | + (−0.341 + 0.341i)5-s + 0.505·7-s + (−0.590 − 0.590i)11-s + (−1.16 + 1.16i)13-s − 0.978i·17-s + (0.979 + 0.979i)19-s + 1.84i·23-s + 0.766i·25-s + (−0.230 − 0.230i)29-s + 0.516i·31-s + (−0.172 + 0.172i)35-s + (−0.0714 − 0.0714i)37-s − 1.21·41-s + (0.838 − 0.838i)43-s − 0.467·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9992437442\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9992437442\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.763 - 0.763i)T - 5iT^{2} \) |
| 7 | \( 1 - 1.33T + 7T^{2} \) |
| 11 | \( 1 + (1.95 + 1.95i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.18 - 4.18i)T - 13iT^{2} \) |
| 17 | \( 1 + 4.03iT - 17T^{2} \) |
| 19 | \( 1 + (-4.26 - 4.26i)T + 19iT^{2} \) |
| 23 | \( 1 - 8.86iT - 23T^{2} \) |
| 29 | \( 1 + (1.23 + 1.23i)T + 29iT^{2} \) |
| 31 | \( 1 - 2.87iT - 31T^{2} \) |
| 37 | \( 1 + (0.434 + 0.434i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.81T + 41T^{2} \) |
| 43 | \( 1 + (-5.49 + 5.49i)T - 43iT^{2} \) |
| 47 | \( 1 + 3.20T + 47T^{2} \) |
| 53 | \( 1 + (4.06 - 4.06i)T - 53iT^{2} \) |
| 59 | \( 1 + (-4.71 - 4.71i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.26 - 3.26i)T - 61iT^{2} \) |
| 67 | \( 1 + (-5.44 - 5.44i)T + 67iT^{2} \) |
| 71 | \( 1 + 3.76iT - 71T^{2} \) |
| 73 | \( 1 - 10.5iT - 73T^{2} \) |
| 79 | \( 1 - 11.1iT - 79T^{2} \) |
| 83 | \( 1 + (9.73 - 9.73i)T - 83iT^{2} \) |
| 89 | \( 1 + 1.64T + 89T^{2} \) |
| 97 | \( 1 + 5.70T + 97T^{2} \) |
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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
−9.864010684409221063300311332557, −9.413843514851316144714445339052, −8.308873933845171591757322617280, −7.41061353487868000477387627939, −7.07996656646952993046829877247, −5.60413036528285178583744610605, −5.05965771545045169108669779309, −3.84106097670396870949317590115, −2.89193578401666380586860720634, −1.58226464185772586945224340058,
0.42769768670912195208352153951, 2.14346257923177561722377646338, 3.17068063529767704388523370524, 4.68622913977101672689098319580, 4.91913277374457810474400451757, 6.16328904967932361721787813284, 7.24022468974718527231792374264, 7.972835094962028674388608849658, 8.510573617483947363617159980854, 9.685362538800088505006037231414