L(s) = 1 | + i·2-s + (−1.15 − 1.15i)3-s − 4-s + (1.65 − 1.5i)5-s + (1.15 − 1.15i)6-s + 3·7-s − i·8-s − 0.316i·9-s + (1.5 + 1.65i)10-s + (3.31 + 3.31i)11-s + (1.15 + 1.15i)12-s + (−3 − 2i)13-s + 3i·14-s + (−3.65 − 0.183i)15-s + 16-s + (−3.15 − 3.15i)17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.668 − 0.668i)3-s − 0.5·4-s + (0.741 − 0.670i)5-s + (0.472 − 0.472i)6-s + 1.13·7-s − 0.353i·8-s − 0.105i·9-s + (0.474 + 0.524i)10-s + (1.00 + 1.00i)11-s + (0.334 + 0.334i)12-s + (−0.832 − 0.554i)13-s + 0.801i·14-s + (−0.944 − 0.0473i)15-s + 0.250·16-s + (−0.766 − 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01589 - 0.0570826i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01589 - 0.0570826i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (-1.65 + 1.5i)T \) |
| 13 | \( 1 + (3 + 2i)T \) |
good | 3 | \( 1 + (1.15 + 1.15i)T + 3iT^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + (-3.31 - 3.31i)T + 11iT^{2} \) |
| 17 | \( 1 + (3.15 + 3.15i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2 - 2i)T + 19iT^{2} \) |
| 23 | \( 1 + (3.31 - 3.31i)T - 23iT^{2} \) |
| 29 | \( 1 - 6.31iT - 29T^{2} \) |
| 31 | \( 1 + (-1 + i)T - 31iT^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + (6.31 - 6.31i)T - 41iT^{2} \) |
| 43 | \( 1 + (-2.47 + 2.47i)T - 43iT^{2} \) |
| 47 | \( 1 + 9.31T + 47T^{2} \) |
| 53 | \( 1 + (-9.63 - 9.63i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.316 + 0.316i)T - 59iT^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4.94iT - 67T^{2} \) |
| 71 | \( 1 + (2.84 - 2.84i)T - 71iT^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 12.9iT - 79T^{2} \) |
| 83 | \( 1 - 6.31T + 83T^{2} \) |
| 89 | \( 1 + (-0.316 + 0.316i)T - 89iT^{2} \) |
| 97 | \( 1 + 8.94iT - 97T^{2} \) |
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
−13.29898795407723739480962772167, −12.27496266241979411528259645129, −11.63576426191260636346735977461, −9.943878323259637616898362751324, −9.024943728555946670271473252843, −7.68959152861224254696584351895, −6.72835072096288243517597233164, −5.51601220307747301594781447428, −4.61301069285376201831877606548, −1.52685583126304549845630754145,
2.12069995446381468213987645657, 4.11231289958349244744659309402, 5.25376304505927525895188092657, 6.49379453968045549775435707444, 8.261048674711215591901401423258, 9.507378864288517136660624141310, 10.45776097518357187206765390509, 11.27797645355773336438294468932, 11.77855932080962877575641752925, 13.46921300489278315036024924363