| L(s) = 1 | + 3i·3-s + i·7-s − 6·9-s − 5·11-s − 3i·13-s + i·17-s − 6·19-s − 3·21-s + 6i·23-s − 9i·27-s + 9·29-s − 4·31-s − 15i·33-s − 2i·37-s + 9·39-s + ⋯ |
| L(s) = 1 | + 1.73i·3-s + 0.377i·7-s − 2·9-s − 1.50·11-s − 0.832i·13-s + 0.242i·17-s − 1.37·19-s − 0.654·21-s + 1.25i·23-s − 1.73i·27-s + 1.67·29-s − 0.718·31-s − 2.61i·33-s − 0.328i·37-s + 1.44·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.156387 - 0.662467i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.156387 - 0.662467i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 - 3iT - 3T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - iT - 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 13T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 4T + 89T^{2} \) |
| 97 | \( 1 - 13iT - 97T^{2} \) |
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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
−10.59488606638497802590471713852, −10.28075442652587007118086495539, −9.372014338397061812541504654221, −8.489742183690883841737694427104, −7.80716655481365557359450062159, −6.16273023388888876329538437111, −5.29922882007952681476046861828, −4.66219609678196210872593875525, −3.49254032821134777410237607631, −2.58645479323628559486467999830,
0.33054285764930520896086152591, 1.93571542801500409457149809263, 2.77702407396729899599472009502, 4.49095532128702112856630399602, 5.68003026494937840684664637836, 6.73477479928210766509869206677, 7.11136986237823010309136103811, 8.290200500641146864512453704002, 8.554091312818740045830200041330, 10.15362406494797416618628475084
