| L(s) = 1 | + i·2-s − 4-s + 2i·7-s − i·8-s − i·13-s − 2·14-s + 16-s − 2·19-s − 6i·23-s + 26-s − 2i·28-s + 8·31-s + i·32-s + 2i·37-s − 2i·38-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.755i·7-s − 0.353i·8-s − 0.277i·13-s − 0.534·14-s + 0.250·16-s − 0.458·19-s − 1.25i·23-s + 0.196·26-s − 0.377i·28-s + 1.43·31-s + 0.176i·32-s + 0.328i·37-s − 0.324i·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.693957128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693957128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 4iT - 97T^{2} \) |
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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
−8.357612652483960015748794882807, −7.56867273846406347520545378260, −6.52774847602070219678404972356, −6.36189709835583103507743310380, −5.34043124017929858160796901315, −4.81387431686754643202459429034, −3.95220834044452223976704913190, −2.94450213085990529260451166360, −2.11663525527799189878603624329, −0.70478057866364163643810597504,
0.68392158137403496962316249758, 1.66554726581324090051367487739, 2.61015650266213153172502531449, 3.57821501861822230040911078706, 4.13348682736068882176950817396, 4.93869346635543516892796418090, 5.72352358436234693252106864602, 6.64857283899997342031286431185, 7.28613287940169713603245650360, 8.081464479895218380636089235908
