L(s) = 1 | + (−0.707 + 0.707i)3-s + (−2.12 + 0.707i)5-s + (−1 − i)7-s − 1.00i·9-s − 5.41i·11-s + (3.41 + 3.41i)13-s + (0.999 − 2i)15-s + (5.41 − 5.41i)17-s − 4.82·19-s + 1.41·21-s + (4.24 − 4.24i)23-s + (3.99 − 3i)25-s + (0.707 + 0.707i)27-s − 0.242i·29-s + 1.65i·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (−0.948 + 0.316i)5-s + (−0.377 − 0.377i)7-s − 0.333i·9-s − 1.63i·11-s + (0.946 + 0.946i)13-s + (0.258 − 0.516i)15-s + (1.31 − 1.31i)17-s − 1.10·19-s + 0.308·21-s + (0.884 − 0.884i)23-s + (0.799 − 0.600i)25-s + (0.136 + 0.136i)27-s − 0.0450i·29-s + 0.297i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.749877 - 0.418084i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.749877 - 0.418084i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.12 - 0.707i)T \) |
good | 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.41iT - 11T^{2} \) |
| 13 | \( 1 + (-3.41 - 3.41i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5.41 + 5.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 + (-4.24 + 4.24i)T - 23iT^{2} \) |
| 29 | \( 1 + 0.242iT - 29T^{2} \) |
| 31 | \( 1 - 1.65iT - 31T^{2} \) |
| 37 | \( 1 + (0.585 - 0.585i)T - 37iT^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 + (-4.82 + 4.82i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.24 + 6.24i)T + 47iT^{2} \) |
| 53 | \( 1 + (2.82 + 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 - 1.41T + 59T^{2} \) |
| 61 | \( 1 + 3.17T + 61T^{2} \) |
| 67 | \( 1 + (3.65 + 3.65i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.34iT - 71T^{2} \) |
| 73 | \( 1 + (-7.48 - 7.48i)T + 73iT^{2} \) |
| 79 | \( 1 - 2.34T + 79T^{2} \) |
| 83 | \( 1 + (-3.07 + 3.07i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.65iT - 89T^{2} \) |
| 97 | \( 1 + (-0.656 + 0.656i)T - 97iT^{2} \) |
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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
−10.99837834358854903281929938990, −10.18334706887135345321171844250, −8.923964694202513748391421009786, −8.299897228001785872597828110557, −7.00344904746531792118800649766, −6.31679007014803612186606192828, −5.07243960438305366804522719080, −3.85543886400834405175648274367, −3.15786608490863518431817566907, −0.60812656207555306067085475376,
1.45281066946860217829463744080, 3.26349678920619026469348160229, 4.40229502558465218870390475151, 5.53972015301862097837353160932, 6.52242469678607630504753102602, 7.63555118206943388984571425302, 8.185882915649647404024575183876, 9.341353913055162965908657706357, 10.40891539312751648789035870792, 11.13976385482226363130461554817