| L(s) = 1 | + (−1.49 − 0.683i)2-s + (0.900 − 3.06i)3-s + (0.465 + 0.536i)4-s + (2.15 − 0.592i)5-s + (−3.44 + 3.97i)6-s + (0.632 + 0.0908i)7-s + (0.598 + 2.03i)8-s + (−6.07 − 3.90i)9-s + (−3.63 − 0.587i)10-s + (1.69 + 3.71i)11-s + (2.06 − 0.943i)12-s + (−3.19 + 0.459i)13-s + (−0.884 − 0.568i)14-s + (0.124 − 7.14i)15-s + (0.699 − 4.86i)16-s + (0.965 + 0.836i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 0.483i)2-s + (0.519 − 1.77i)3-s + (0.232 + 0.268i)4-s + (0.964 − 0.264i)5-s + (−1.40 + 1.62i)6-s + (0.238 + 0.0343i)7-s + (0.211 + 0.720i)8-s + (−2.02 − 1.30i)9-s + (−1.14 − 0.185i)10-s + (0.512 + 1.12i)11-s + (0.596 − 0.272i)12-s + (−0.885 + 0.127i)13-s + (−0.236 − 0.151i)14-s + (0.0321 − 1.84i)15-s + (0.174 − 1.21i)16-s + (0.234 + 0.202i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.339266 - 0.719910i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.339266 - 0.719910i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.15 + 0.592i)T \) |
| 23 | \( 1 + (-3.19 - 3.57i)T \) |
| good | 2 | \( 1 + (1.49 + 0.683i)T + (1.30 + 1.51i)T^{2} \) |
| 3 | \( 1 + (-0.900 + 3.06i)T + (-2.52 - 1.62i)T^{2} \) |
| 7 | \( 1 + (-0.632 - 0.0908i)T + (6.71 + 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.69 - 3.71i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (3.19 - 0.459i)T + (12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (-0.965 - 0.836i)T + (2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (-0.307 - 0.354i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (2.61 - 3.02i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (4.96 - 1.45i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-5.16 + 8.04i)T + (-15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (-7.36 + 4.73i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-0.206 + 0.704i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 - 0.589iT - 47T^{2} \) |
| 53 | \( 1 + (6.64 + 0.955i)T + (50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (0.734 + 5.11i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (2.44 - 0.717i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-2.18 - 0.998i)T + (43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (3.46 - 7.58i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-0.557 + 0.483i)T + (10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.33 - 16.2i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (5.64 - 8.77i)T + (-34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (-5.75 - 1.68i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (3.48 + 5.42i)T + (-40.2 + 88.2i)T^{2} \) |
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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
−12.93905283962126364623015835344, −12.35290068434062816070872690623, −11.13809201430257347337859487974, −9.594952872523033482286026032486, −9.064048877106030051696732661109, −7.79522211001747255529242700499, −6.97807790102531793396291491630, −5.44452462967791102468656379124, −2.33056250606970140160981838662, −1.44939175523452243883655671941,
3.09681230219379547452375467894, 4.66597763105876990766619099704, 6.10214581702670664620889491736, 7.85456077979324100553929796816, 9.002929484738144262166823546756, 9.485840265644756438972729403885, 10.34153445936891190186414227565, 11.21796728347416544546311500022, 13.25996939221660031453445543683, 14.39994331825694404158058378783
