L(s) = 1 | + (−1.40 + 0.178i)2-s + (1.93 − 0.5i)4-s + (1.58 − 1.58i)5-s + (−2.44 − 2.44i)7-s + (−2.62 + 1.04i)8-s − 3i·9-s + (−1.93 + 2.50i)10-s − 2.44·11-s + (−0.418 + 3.58i)13-s + (3.87 + 3i)14-s + (3.50 − 1.93i)16-s + (−1 − i)17-s + (0.534 + 4.20i)18-s − 2.44i·19-s + (2.27 − 3.85i)20-s + ⋯ |
L(s) = 1 | + (−0.992 + 0.126i)2-s + (0.968 − 0.250i)4-s + (0.707 − 0.707i)5-s + (−0.925 − 0.925i)7-s + (−0.929 + 0.370i)8-s − i·9-s + (−0.612 + 0.790i)10-s − 0.738·11-s + (−0.116 + 0.993i)13-s + (1.03 + 0.801i)14-s + (0.875 − 0.484i)16-s + (−0.242 − 0.242i)17-s + (0.126 + 0.992i)18-s − 0.561i·19-s + (0.507 − 0.861i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0954 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0954 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.465519 - 0.512283i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.465519 - 0.512283i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.40 - 0.178i)T \) |
| 5 | \( 1 + (-1.58 + 1.58i)T \) |
| 13 | \( 1 + (0.418 - 3.58i)T \) |
good | 3 | \( 1 + 3iT^{2} \) |
| 7 | \( 1 + (2.44 + 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.44T + 11T^{2} \) |
| 17 | \( 1 + (1 + i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 + (3.87 + 3.87i)T + 23iT^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 - 9.79T + 31T^{2} \) |
| 37 | \( 1 + (-6.32 + 6.32i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.32iT - 41T^{2} \) |
| 43 | \( 1 + (-7.74 - 7.74i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.44 + 2.44i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2 + 2i)T - 53iT^{2} \) |
| 59 | \( 1 - 7.34iT - 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (2.44 + 2.44i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 + (-3.16 - 3.16i)T + 73iT^{2} \) |
| 79 | \( 1 - 7.74T + 79T^{2} \) |
| 83 | \( 1 + (-4.89 + 4.89i)T - 83iT^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 + (3.16 - 3.16i)T - 97iT^{2} \) |
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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
−11.64200620419384512786022384883, −10.41244465441859136278962688294, −9.675611340567036095337112896718, −9.123381186125612196780184108549, −7.936600918089648935411193895418, −6.68214513258509052095196989994, −6.13212236835594186534416572164, −4.38704614279648199788569428496, −2.60328408124032007000203627933, −0.71171634099338401619364074715,
2.23800383597365298541901517961, 3.06272147379643403282507543331, 5.54566660622598963055041655942, 6.25791343589986655750641630295, 7.54449945127375168841017713718, 8.369439894834561498815855329381, 9.620303608465968461601414720643, 10.21297309053889780671626399760, 10.91483896455749007256308946631, 12.14471666378307717964540038198