| L(s) = 1 | + (−1.61 − 1.93i)2-s + (0.0453 + 0.257i)3-s + (−0.755 + 4.28i)4-s + (4.11 − 0.726i)5-s + (0.422 − 0.503i)6-s + (−0.270 + 2.63i)7-s + (5.13 − 2.96i)8-s + (2.75 − 1.00i)9-s + (−8.07 − 6.77i)10-s − 1.33·11-s − 1.13·12-s + (−1.68 − 1.41i)13-s + (5.51 − 3.74i)14-s + (0.373 + 1.02i)15-s + (−5.85 − 2.13i)16-s + (−0.498 + 1.36i)17-s + ⋯ |
| L(s) = 1 | + (−1.14 − 1.36i)2-s + (0.0261 + 0.148i)3-s + (−0.377 + 2.14i)4-s + (1.84 − 0.324i)5-s + (0.172 − 0.205i)6-s + (−0.102 + 0.994i)7-s + (1.81 − 1.04i)8-s + (0.918 − 0.334i)9-s + (−2.55 − 2.14i)10-s − 0.401·11-s − 0.327·12-s + (−0.467 − 0.392i)13-s + (1.47 − 1.00i)14-s + (0.0963 + 0.264i)15-s + (−1.46 − 0.532i)16-s + (−0.120 + 0.332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.658440 - 0.463205i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.658440 - 0.463205i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.270 - 2.63i)T \) |
| 19 | \( 1 + (2.62 + 3.48i)T \) |
| good | 2 | \( 1 + (1.61 + 1.93i)T + (-0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.0453 - 0.257i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (-4.11 + 0.726i)T + (4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + 1.33T + 11T^{2} \) |
| 13 | \( 1 + (1.68 + 1.41i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.498 - 1.36i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (1.69 + 1.42i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.87 - 0.683i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.09 - 3.63i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.39 - 4.26i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0190 + 0.0159i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (7.24 + 2.63i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (0.863 + 2.37i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (9.18 + 1.61i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-8.28 - 3.01i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (6.85 - 8.16i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (2.99 - 3.56i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.908 + 2.49i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (0.160 - 0.0282i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-3.69 + 10.1i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-5.12 - 2.96i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.59 - 9.06i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.30 - 7.38i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−12.76085394323939740635411282445, −12.09300606825204430245459698058, −10.48355557184947050812543354212, −10.06618516956660284961361702918, −9.181762625924085951218486170757, −8.476934388373962159191850656677, −6.57146624392185283277500031787, −5.00807855873724045309578280653, −2.79126255444835188991261756871, −1.72187799698146988247331112623,
1.73232159504208496463150218654, 4.96509410622470612226118738862, 6.25915440767725478548977984814, 6.93581572565487216102079444881, 7.968591574509083758701687400474, 9.417467021067098041658324718924, 10.04939201743990261055609827834, 10.58760694476160879563866213322, 12.92689302727215221994878522391, 13.88777482260147549847144998212
