L(s) = 1 | + i·2-s − 2i·3-s + 4-s + (−2 + i)5-s + 2·6-s + 3i·8-s − 9-s + (−1 − 2i)10-s + 2·11-s − 2i·12-s + (2 + 4i)15-s − 16-s − i·18-s + 6·19-s + (−2 + i)20-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.15i·3-s + 0.5·4-s + (−0.894 + 0.447i)5-s + 0.816·6-s + 1.06i·8-s − 0.333·9-s + (−0.316 − 0.632i)10-s + 0.603·11-s − 0.577i·12-s + (0.516 + 1.03i)15-s − 0.250·16-s − 0.235i·18-s + 1.37·19-s + (−0.447 + 0.223i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67694 + 0.395873i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67694 + 0.395873i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2 - i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - iT - 2T^{2} \) |
| 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 - 2T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 8T + 89T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−10.39738332494281773015115349211, −9.135028408448085048348604819221, −8.070529485534871437451252931988, −7.47854159985862510342277129721, −7.02035157583080860657376005505, −6.26229591178385040676011346133, −5.25619461281874245704407629725, −3.77277384520025032372025759228, −2.61428170700800181089435165401, −1.25492364593364328541105446409,
1.05431569957723382609476816605, 2.79451815074023346142904768737, 3.79506898031740760204861998158, 4.35408006724149895531059745780, 5.46172662286636619768773705917, 6.78066107539005019369071228596, 7.59537686490901540053591066642, 8.742205774272296072533984301761, 9.504787394183467055495503301698, 10.18807562390963600654414739442