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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.07221210754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07221210754\) |
\(L(\frac{3}{2})\) |
|
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
−10.16894324182621056985339605752, −9.386405380965716226246133008000, −8.381505975277154451165649537920, −7.88700670622452234527953711376, −6.80808322788832924932148117483, −5.92866418673352794308930916273, −5.11468300912744510458023865225, −3.89934709385724621517871477307, −3.18957371884096940236726434643, −1.66333261252638802344417422557,
0.02972995367577364786695270918, 2.06122612467356578219810420971, 3.02690293473777512933727786408, 4.36034410590887360138281993186, 4.83229827425753610258315845871, 6.43593446487020116993629800103, 6.83437016304821172721642834637, 7.59713922296463744278366473588, 8.824564038160072887812459576374, 9.409934392621702022704252093736