| L(s) = 1 | + (0.5 + 0.866i)3-s + (−1.70 + 2.95i)5-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + 2.58·13-s − 3.41·15-s + (1.12 + 1.94i)17-s + (−1.41 + 2.44i)19-s + (−3.82 + 6.63i)23-s + (−3.32 − 5.76i)25-s − 0.999·27-s − 6.82·29-s + (−0.585 − 1.01i)31-s + (0.999 − 1.73i)33-s + (2 − 3.46i)37-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.763 + 1.32i)5-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + 0.717·13-s − 0.881·15-s + (0.271 + 0.471i)17-s + (−0.324 + 0.561i)19-s + (−0.798 + 1.38i)23-s + (−0.665 − 1.15i)25-s − 0.192·27-s − 1.26·29-s + (−0.105 − 0.182i)31-s + (0.174 − 0.301i)33-s + (0.328 − 0.569i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6834253136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6834253136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (1.70 - 2.95i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 + (-1.12 - 1.94i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.82 - 6.63i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 + (0.585 + 1.01i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.585 + 1.01i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.12 - 10.6i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.82 + 4.89i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 9.31T + 71T^{2} \) |
| 73 | \( 1 + (6.94 + 12.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.82 + 11.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7.31T + 83T^{2} \) |
| 89 | \( 1 + (-7.12 + 12.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.58T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.437566814394664292374963045447, −8.609248533025890810441998619930, −7.67301407763802092815499323230, −7.47509567713771011945610394565, −6.12828046890969184957813607322, −5.77608615893460746556427438818, −4.33370124531380089695385918679, −3.57813896172086589162832373592, −3.13446574965654026121654987078, −1.84515130578861641669598011811,
0.22519150083049940317497307569, 1.36358588209344837365117866681, 2.51973511172944481287662037351, 3.76603456855653435810253070265, 4.49222365447608573408195898954, 5.26464166278851014294869468191, 6.25501695047126728192532491829, 7.16653593212079323796056702970, 7.948579431692616415563814484343, 8.449219698459344939756476535867
