| 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
−11.54508952116823658170718024842, −10.93969799470002079717158806909, −10.30459275355908297124219039775, −9.097672296286504505739664402863, −8.019066687057263545510728266879, −7.15420647912561557503599871188, −6.35009846553633379097506446708, −4.47983331336878564466369264759, −2.47343052114957686509362929410, −0.48998198338046678892037770887,
1.21766169777927125396581284609, 4.04227161965002617208355037170, 5.46328705354978341723206500079, 6.76575527452397781885724668147, 7.57258222774442878992314940247, 8.769223892391147790476814723697, 9.348415101934838874036554042825, 10.71334145010124179753064570172, 11.02520749983099396319188227429, 12.17686165636652656293430888615
