L(s) = 1 | + 2.69i·2-s − i·3-s − 5.27·4-s − 3.60i·5-s + 2.69·6-s − 8.81i·8-s − 9-s + 9.72·10-s + (−2.32 − 2.36i)11-s + 5.27i·12-s + 4.29·13-s − 3.60·15-s + 13.2·16-s − 6.77·17-s − 2.69i·18-s + 2.03·19-s + ⋯ |
L(s) = 1 | + 1.90i·2-s − 0.577i·3-s − 2.63·4-s − 1.61i·5-s + 1.10·6-s − 3.11i·8-s − 0.333·9-s + 3.07·10-s + (−0.699 − 0.714i)11-s + 1.52i·12-s + 1.19·13-s − 0.931·15-s + 3.30·16-s − 1.64·17-s − 0.635i·18-s + 0.465·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2716464002\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2716464002\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (2.32 + 2.36i)T \) |
good | 2 | \( 1 - 2.69iT - 2T^{2} \) |
| 5 | \( 1 + 3.60iT - 5T^{2} \) |
| 13 | \( 1 - 4.29T + 13T^{2} \) |
| 17 | \( 1 + 6.77T + 17T^{2} \) |
| 19 | \( 1 - 2.03T + 19T^{2} \) |
| 23 | \( 1 - 1.20T + 23T^{2} \) |
| 29 | \( 1 - 0.635iT - 29T^{2} \) |
| 31 | \( 1 - 5.92iT - 31T^{2} \) |
| 37 | \( 1 + 9.93T + 37T^{2} \) |
| 41 | \( 1 + 1.61T + 41T^{2} \) |
| 43 | \( 1 - 6.25iT - 43T^{2} \) |
| 47 | \( 1 - 0.262iT - 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 + 3.79iT - 59T^{2} \) |
| 61 | \( 1 + 6.15T + 61T^{2} \) |
| 67 | \( 1 - 6.29T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 5.55T + 73T^{2} \) |
| 79 | \( 1 + 12.5iT - 79T^{2} \) |
| 83 | \( 1 - 1.67T + 83T^{2} \) |
| 89 | \( 1 - 8.75iT - 89T^{2} \) |
| 97 | \( 1 + 2.61iT - 97T^{2} \) |
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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
−8.832547368996987761369234174151, −8.329644328434269345412442657245, −7.64177781315286869336949575545, −6.63326107436819484242606873716, −6.03130856387672913268202831161, −5.15692367611745614669366126531, −4.70606946840557594080286468231, −3.53451962317713578178456487368, −1.32257559596447586422923106910, −0.11173572224133136144972443325,
1.91787009822172972053565836835, 2.68384560143045458937566934843, 3.48412546148982778942792499439, 4.17607946596231524732187048019, 5.15163220123799145054934721444, 6.28215443613085814605702538449, 7.37130041857328238216212908483, 8.499259738438932632949184627058, 9.189995881543039382144945074827, 10.08254940026568245601790616611