L(s) = 1 | + (1.28 + 1.16i)3-s − 0.936·5-s + 3.88i·7-s + (0.280 + 2.98i)9-s − 4.27·11-s + 2.56·13-s + (−1.19 − 1.09i)15-s + 7.60·17-s − 6.07i·19-s + (−4.53 + 4.98i)21-s + (2.39 + 4.15i)23-s − 4.12·25-s + (−3.12 + 4.15i)27-s − 5.97i·29-s + 3.68·31-s + ⋯ |
L(s) = 1 | + (0.739 + 0.673i)3-s − 0.418·5-s + 1.46i·7-s + (0.0935 + 0.995i)9-s − 1.28·11-s + 0.710·13-s + (−0.309 − 0.281i)15-s + 1.84·17-s − 1.39i·19-s + (−0.989 + 1.08i)21-s + (0.500 + 0.865i)23-s − 0.824·25-s + (−0.601 + 0.799i)27-s − 1.10i·29-s + 0.661·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13039 + 0.910425i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13039 + 0.910425i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.28 - 1.16i)T \) |
| 23 | \( 1 + (-2.39 - 4.15i)T \) |
good | 5 | \( 1 + 0.936T + 5T^{2} \) |
| 7 | \( 1 - 3.88iT - 7T^{2} \) |
| 11 | \( 1 + 4.27T + 11T^{2} \) |
| 13 | \( 1 - 2.56T + 13T^{2} \) |
| 17 | \( 1 - 7.60T + 17T^{2} \) |
| 19 | \( 1 + 6.07iT - 19T^{2} \) |
| 29 | \( 1 + 5.97iT - 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 - 2.18iT - 37T^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 - 6.07iT - 43T^{2} \) |
| 47 | \( 1 - 1.30iT - 47T^{2} \) |
| 53 | \( 1 - 8.13T + 53T^{2} \) |
| 59 | \( 1 + 4.66iT - 59T^{2} \) |
| 61 | \( 1 + 7.77iT - 61T^{2} \) |
| 67 | \( 1 + 11.6iT - 67T^{2} \) |
| 71 | \( 1 - 2.33iT - 71T^{2} \) |
| 73 | \( 1 + 5.68T + 73T^{2} \) |
| 79 | \( 1 - 1.70iT - 79T^{2} \) |
| 83 | \( 1 + 0.525T + 83T^{2} \) |
| 89 | \( 1 + 1.46T + 89T^{2} \) |
| 97 | \( 1 + 9.96iT - 97T^{2} \) |
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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
−12.00255254706708102885420401805, −11.14721491516759808786925703332, −10.03803427076643348129315859025, −9.214646234199610983491166817733, −8.271773050427809451585667776041, −7.62232071269339407621228132160, −5.77057205288381172401598373208, −4.99429887855410325458190111892, −3.43429768865754316565780590496, −2.46519556850254437040356775606,
1.15459944877248117764411251767, 3.12225864317805090074164937788, 4.05276797566720825448117518440, 5.73296833877342311546368726279, 7.11359717857339011688054583829, 7.79676824116551012530971232427, 8.412229809525462587466596449058, 10.01956751287470982331840656759, 10.49918071287841470736730056694, 11.85713573266385480957527303495