L(s) = 1 | + (−0.178 + 1.40i)2-s + (−1.93 − 0.5i)4-s + (1.58 − 1.58i)5-s + (2.44 + 2.44i)7-s + (1.04 − 2.62i)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 + (4.20 + 0.534i)18-s + 2.44i·19-s + (−3.85 + 2.27i)20-s + ⋯ |
L(s) = 1 | + (−0.126 + 0.992i)2-s + (−0.968 − 0.250i)4-s + (0.707 − 0.707i)5-s + (0.925 + 0.925i)7-s + (0.370 − 0.929i)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 + (0.992 + 0.126i)18-s + 0.561i·19-s + (−0.861 + 0.507i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17357 + 0.618349i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17357 + 0.618349i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.178 - 1.40i)T \) |
| 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
−12.24122099317061591450490047475, −11.35557026957247131310864018682, −9.591643391292293954800961368064, −9.177727662264025988721555931437, −8.451839188183648529874603915748, −7.09086269540284467417999595148, −6.03064305937472311179600113266, −5.22677961236133929375591998199, −4.05977718088841325333421298362, −1.59227403407412329797455187407,
1.54820177126435894202870823023, 2.91026128516531628028517714482, 4.38676027525123866439003553849, 5.41244736634111049401996388134, 7.08016801492524016624396240964, 8.051311521455995268834692773103, 9.166600130951731187719085524977, 10.38063550355680938713603827306, 10.76294749324272330517961577360, 11.46971687135094154311329229558