L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.809 − 1.40i)5-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (0.809 + 1.40i)10-s + (−0.309 − 0.535i)13-s + (−0.5 + 0.866i)16-s + (−0.309 + 0.535i)17-s + 0.999·18-s − 1.61·20-s + (−0.809 − 1.40i)25-s + 0.618·26-s + (−0.309 − 0.535i)29-s + (−0.499 − 0.866i)32-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.809 − 1.40i)5-s + 0.999·8-s + (−0.5 − 0.866i)9-s + (0.809 + 1.40i)10-s + (−0.309 − 0.535i)13-s + (−0.5 + 0.866i)16-s + (−0.309 + 0.535i)17-s + 0.999·18-s − 1.61·20-s + (−0.809 − 1.40i)25-s + 0.618·26-s + (−0.309 − 0.535i)29-s + (−0.499 − 0.866i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8075912998\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8075912998\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 1.61T + T^{2} \) |
| 41 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−9.215901875194306133031509945630, −8.937916788717833124574499501562, −8.193992782758745333592010595344, −7.23194614467255245420540359089, −6.15819531598493370851108556594, −5.64709729268612823645904710603, −4.90999343305934473845420631102, −3.86314930709287956649059214821, −2.06607959945994590687584689633, −0.78127675763070228642618771759,
1.90919156669832866666906432131, 2.57999526727318222766026317581, 3.42203369604030025271906319742, 4.70422006356188020422658812839, 5.68921742718386917897181156451, 6.86292690294591347329559660141, 7.36334336426060933337947603882, 8.422872852838874027877214753024, 9.225132979344759879161823411930, 10.03848423497485658477556538856