L(s) = 1 | − 2.76i·5-s − i·7-s − 2.55·11-s − 2.32·13-s − 7.36i·17-s − 1.78i·19-s − 3.87·23-s − 2.64·25-s − 7.32i·29-s − 2.17i·31-s − 2.76·35-s − 1.24·37-s + 3.52i·41-s + 5.28i·43-s + 5.77·47-s + ⋯ |
L(s) = 1 | − 1.23i·5-s − 0.377i·7-s − 0.770·11-s − 0.645·13-s − 1.78i·17-s − 0.408i·19-s − 0.807·23-s − 0.529·25-s − 1.36i·29-s − 0.390i·31-s − 0.467·35-s − 0.205·37-s + 0.550i·41-s + 0.806i·43-s + 0.842·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5563817764\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5563817764\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 2.76iT - 5T^{2} \) |
| 11 | \( 1 + 2.55T + 11T^{2} \) |
| 13 | \( 1 + 2.32T + 13T^{2} \) |
| 17 | \( 1 + 7.36iT - 17T^{2} \) |
| 19 | \( 1 + 1.78iT - 19T^{2} \) |
| 23 | \( 1 + 3.87T + 23T^{2} \) |
| 29 | \( 1 + 7.32iT - 29T^{2} \) |
| 31 | \( 1 + 2.17iT - 31T^{2} \) |
| 37 | \( 1 + 1.24T + 37T^{2} \) |
| 41 | \( 1 - 3.52iT - 41T^{2} \) |
| 43 | \( 1 - 5.28iT - 43T^{2} \) |
| 47 | \( 1 - 5.77T + 47T^{2} \) |
| 53 | \( 1 - 7.29iT - 53T^{2} \) |
| 59 | \( 1 + 4.98T + 59T^{2} \) |
| 61 | \( 1 - 3.15T + 61T^{2} \) |
| 67 | \( 1 - 7.72iT - 67T^{2} \) |
| 71 | \( 1 - 7.98T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 2.10iT - 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 9.71iT - 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−7.61880198646234648937513180344, −7.16904011213309385161997447834, −6.06329829463602665860688743028, −5.36891311498259499294340638913, −4.65613709229020026510389871769, −4.29158599514658549193636094399, −2.97283020775687669178406960095, −2.26388292764439330377993565633, −0.976385572331423173464086848941, −0.15436328805167811404023252686,
1.72998757300110511806961548387, 2.44416066271502776411794347877, 3.31097480477827163573240852619, 3.93938611102401915592117431909, 5.06088341026894070337993849206, 5.73827534543079085424940716865, 6.43098079361527636258619447841, 7.08134484704226988647472904368, 7.77399456842273917743324761309, 8.401674786169622199224671904158