L(s) = 1 | − i·2-s + 2i·3-s − 4-s + 2·6-s + i·8-s − 9-s + 4·11-s − 2i·12-s − 2i·13-s + 16-s − 8i·17-s + i·18-s + 5·19-s − 4i·22-s + i·23-s − 2·24-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.15i·3-s − 0.5·4-s + 0.816·6-s + 0.353i·8-s − 0.333·9-s + 1.20·11-s − 0.577i·12-s − 0.554i·13-s + 0.250·16-s − 1.94i·17-s + 0.235i·18-s + 1.14·19-s − 0.852i·22-s + 0.208i·23-s − 0.408·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.783853306\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.783853306\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 37 | \( 1 + iT \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 + 9iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 3iT - 53T^{2} \) |
| 59 | \( 1 - 11T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 - 3iT - 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−9.353836511598731687283558502368, −8.964739640590789023705026710858, −7.61570405210692853702276394391, −6.95226098665555802628666913861, −5.42452230654057861368683462675, −5.15532546922596569532937510490, −3.84786565396883117942173891020, −3.60392226358927807982553663471, −2.32553874092456749652720793604, −0.823615537290723055001011782252,
1.13245979957493476425247304735, 1.96697823764337574941749382264, 3.62040442589733917764901550062, 4.31318467793165871745320902589, 5.70647041365490591993336614399, 6.20664014723661831520743104943, 6.98691450681450810623797432226, 7.56868881947706170383289424610, 8.297706315180732564783530237341, 9.162980863271723136730212827289