| L(s) = 1 | + (1.48 + 0.316i)2-s + (−0.689 − 1.58i)3-s + (0.287 + 0.127i)4-s + (−0.209 − 2.22i)5-s + (−0.523 − 2.58i)6-s + (0.255 − 0.442i)7-s + (−2.07 − 1.50i)8-s + (−2.04 + 2.19i)9-s + (0.392 − 3.37i)10-s + (0.818 + 0.173i)11-s + (0.00518 − 0.544i)12-s + (2.86 − 0.608i)13-s + (0.520 − 0.577i)14-s + (−3.39 + 1.86i)15-s + (−3.03 − 3.36i)16-s + (2.78 + 2.02i)17-s + ⋯ |
| L(s) = 1 | + (1.05 + 0.223i)2-s + (−0.398 − 0.917i)3-s + (0.143 + 0.0639i)4-s + (−0.0935 − 0.995i)5-s + (−0.213 − 1.05i)6-s + (0.0965 − 0.167i)7-s + (−0.733 − 0.532i)8-s + (−0.683 + 0.730i)9-s + (0.124 − 1.06i)10-s + (0.246 + 0.0524i)11-s + (0.00149 − 0.157i)12-s + (0.793 − 0.168i)13-s + (0.139 − 0.154i)14-s + (−0.876 + 0.482i)15-s + (−0.757 − 0.841i)16-s + (0.676 + 0.491i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22384 - 1.04080i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22384 - 1.04080i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.689 + 1.58i)T \) |
| 5 | \( 1 + (0.209 + 2.22i)T \) |
| good | 2 | \( 1 + (-1.48 - 0.316i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (-0.255 + 0.442i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.818 - 0.173i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-2.86 + 0.608i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.78 - 2.02i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.87 - 3.54i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.89 + 4.32i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.0623 + 0.593i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.0928 + 0.883i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (2.62 - 8.07i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (8.80 - 1.87i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (1.06 - 1.84i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.760 + 7.23i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-7.16 + 5.20i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (7.87 - 1.67i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (2.68 + 0.570i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.160 + 1.52i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (-4.04 + 2.94i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.39 - 10.4i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.788 + 7.50i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-0.410 + 0.182i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (5.39 + 16.6i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.76 - 16.8i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−12.20894949285158335588828684037, −11.69253856874481662626062817203, −10.19495286269067399527738106018, −8.842670162085384940113984358590, −7.938712131750086493584043406030, −6.61653252726460199695239689996, −5.66776526578425085833638756189, −4.86431381819417832085848673332, −3.47659218483026623321135204238, −1.17812970158087793784069664521,
3.04089537167170297036114826390, 3.74325954134625120119749147359, 5.07968300162280584938689482227, 5.88084710220119793989834320248, 7.14073496606109204844271623565, 8.785793339261376680461078925938, 9.672561975255691411920527148714, 10.93605304004724305999056942583, 11.47666502933677835019465408908, 12.25984251192022074028696612721
