| L(s) = 1 | + 2i·2-s − 2·4-s + (−2 + i)5-s − i·7-s + (−2 − 4i)10-s + 2i·13-s + 2·14-s − 4·16-s − 5i·17-s − 8·19-s + (4 − 2i)20-s + i·23-s + (3 − 4i)25-s − 4·26-s + 2i·28-s − 5·29-s + ⋯ |
| L(s) = 1 | + 1.41i·2-s − 4-s + (−0.894 + 0.447i)5-s − 0.377i·7-s + (−0.632 − 1.26i)10-s + 0.554i·13-s + 0.534·14-s − 16-s − 1.21i·17-s − 1.83·19-s + (0.894 − 0.447i)20-s + 0.208i·23-s + (0.600 − 0.800i)25-s − 0.784·26-s + 0.377i·28-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2 - i)T \) |
| 23 | \( 1 - iT \) |
| good | 2 | \( 1 - 2iT - 2T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 5iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 13iT - 67T^{2} \) |
| 71 | \( 1 + 13T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 3iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−9.406450842750462957606546625912, −8.745044866762550126296373013211, −7.79356232252292102221452428941, −7.25938535382878652448366495032, −6.63660284847480759465720970925, −5.71197701573106638805393941302, −4.59189301397514309700714453408, −3.89581783006060886160931008266, −2.38316280791905303307394282737, 0,
1.54203852822146710292753856325, 2.66809820526106136851681215630, 3.82463812313865853962869107329, 4.32087604146460055959991710358, 5.59589557674112292132269255314, 6.71784948448158501661576861377, 7.889808630431927570250696371168, 8.649729408516669510750419551603, 9.290461155938912430628155125580
