| L(s) = 1 | + (−2.27 + 0.484i)2-s + (−1.66 + 0.491i)3-s + (3.13 − 1.39i)4-s + (−1.74 − 1.39i)5-s + (3.54 − 1.92i)6-s + (−0.0399 − 0.0692i)7-s + (−2.69 + 1.95i)8-s + (2.51 − 1.63i)9-s + (4.65 + 2.33i)10-s + (0.117 − 0.0249i)11-s + (−4.51 + 3.85i)12-s + (0.896 + 0.190i)13-s + (0.124 + 0.138i)14-s + (3.58 + 1.45i)15-s + (0.601 − 0.668i)16-s + (−2.65 + 1.92i)17-s + ⋯ |
| L(s) = 1 | + (−1.61 + 0.342i)2-s + (−0.958 + 0.283i)3-s + (1.56 − 0.697i)4-s + (−0.781 − 0.624i)5-s + (1.44 − 0.785i)6-s + (−0.0151 − 0.0261i)7-s + (−0.952 + 0.691i)8-s + (0.838 − 0.544i)9-s + (1.47 + 0.738i)10-s + (0.0353 − 0.00751i)11-s + (−1.30 + 1.11i)12-s + (0.248 + 0.0528i)13-s + (0.0333 + 0.0370i)14-s + (0.926 + 0.376i)15-s + (0.150 − 0.167i)16-s + (−0.643 + 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0718 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0718 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.182426 + 0.196043i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.182426 + 0.196043i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.66 - 0.491i)T \) |
| 5 | \( 1 + (1.74 + 1.39i)T \) |
| good | 2 | \( 1 + (2.27 - 0.484i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (0.0399 + 0.0692i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.117 + 0.0249i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.896 - 0.190i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (2.65 - 1.92i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.09 - 1.51i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-4.80 - 5.33i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.650 - 6.18i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.644 - 6.13i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (3.64 + 11.2i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-6.21 - 1.32i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (-6.13 - 10.6i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.401 + 3.81i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-6.52 - 4.73i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.33 + 0.284i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (9.81 - 2.08i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.885 - 8.42i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (-5.54 - 4.02i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.36 + 4.20i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.223 - 2.12i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-2.75 - 1.22i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (4.33 - 13.3i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (1.27 - 12.1i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−12.17686165636652656293430888615, −11.02520749983099396319188227429, −10.71334145010124179753064570172, −9.348415101934838874036554042825, −8.769223892391147790476814723697, −7.57258222774442878992314940247, −6.76575527452397781885724668147, −5.46328705354978341723206500079, −4.04227161965002617208355037170, −1.21766169777927125396581284609,
0.48998198338046678892037770887, 2.47343052114957686509362929410, 4.47983331336878564466369264759, 6.35009846553633379097506446708, 7.15420647912561557503599871188, 8.019066687057263545510728266879, 9.097672296286504505739664402863, 10.30459275355908297124219039775, 10.93969799470002079717158806909, 11.54508952116823658170718024842
