| L(s) = 1 | + (−1.44 − 0.294i)2-s + (−2.46 + 0.401i)3-s + (0.148 + 0.0631i)4-s + (0.720 − 2.92i)5-s + (3.67 + 0.148i)6-s + (−2.39 + 1.13i)7-s + (2.22 + 1.53i)8-s + (3.09 − 1.03i)9-s + (−1.89 + 3.99i)10-s + (−0.728 + 2.18i)11-s + (−0.391 − 0.0965i)12-s + (1.92 + 3.04i)13-s + (3.78 − 0.932i)14-s + (−0.606 + 7.51i)15-s + (−2.97 − 3.09i)16-s + (1.97 + 4.16i)17-s + ⋯ |
| L(s) = 1 | + (−1.01 − 0.207i)2-s + (−1.42 + 0.231i)3-s + (0.0741 + 0.0315i)4-s + (0.322 − 1.30i)5-s + (1.50 + 0.0604i)6-s + (−0.904 + 0.429i)7-s + (0.786 + 0.542i)8-s + (1.03 − 0.344i)9-s + (−0.600 + 1.26i)10-s + (−0.219 + 0.657i)11-s + (−0.113 − 0.0278i)12-s + (0.534 + 0.845i)13-s + (1.01 − 0.249i)14-s + (−0.156 + 1.93i)15-s + (−0.744 − 0.774i)16-s + (0.479 + 1.01i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.222002 + 0.146195i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.222002 + 0.146195i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.92 - 3.04i)T \) |
| good | 2 | \( 1 + (1.44 + 0.294i)T + (1.83 + 0.783i)T^{2} \) |
| 3 | \( 1 + (2.46 - 0.401i)T + (2.84 - 0.950i)T^{2} \) |
| 5 | \( 1 + (-0.720 + 2.92i)T + (-4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (2.39 - 1.13i)T + (4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (0.728 - 2.18i)T + (-8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (-1.97 - 4.16i)T + (-10.7 + 13.1i)T^{2} \) |
| 19 | \( 1 + (1.79 + 1.03i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.57 - 6.18i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.932 + 4.56i)T + (-26.6 - 11.3i)T^{2} \) |
| 31 | \( 1 + (1.28 + 2.44i)T + (-17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (1.42 - 2.24i)T + (-15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (-1.11 - 6.88i)T + (-38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (7.73 - 4.89i)T + (18.4 - 38.8i)T^{2} \) |
| 47 | \( 1 + (1.71 + 0.208i)T + (45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (7.11 - 10.3i)T + (-18.7 - 49.5i)T^{2} \) |
| 59 | \( 1 + (-8.35 - 8.02i)T + (2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-3.47 + 0.280i)T + (60.2 - 9.78i)T^{2} \) |
| 67 | \( 1 + (5.31 + 12.4i)T + (-46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (-8.65 + 7.06i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-0.619 - 0.699i)T + (-8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (1.68 - 13.8i)T + (-76.7 - 18.9i)T^{2} \) |
| 83 | \( 1 + (-10.5 + 4.00i)T + (62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (7.69 - 4.44i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.44 + 2.73i)T + (81.9 + 51.8i)T^{2} \) |
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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
−12.74934069296589368183990701928, −11.81766695358611231948847576032, −10.86564017545505473404085997164, −9.745821921027399120758646144194, −9.303596215986027229581877421268, −8.131611435298801976528116442574, −6.45873161509662724761289702308, −5.42615482202931518155753945847, −4.46888574247211617584142449066, −1.38198983670162624157366984553,
0.45743614550339639204461641449, 3.31596284240950381315531093039, 5.35915231802218136668157178878, 6.65775049977064124559100888880, 6.98269313766089760147667129572, 8.486376186678968172024393972190, 9.952635410077321135051390843237, 10.55850439891737605857699768611, 11.09986090877292658010492009873, 12.55436560071802421522821959802
