| L(s) = 1 | + (−1.59 + 1.19i)2-s + (−0.271 + 1.32i)3-s + (0.546 − 1.88i)4-s + (−0.978 − 0.513i)5-s + (−1.15 − 2.43i)6-s + (−3.69 − 0.600i)7-s + (−0.0244 − 0.0645i)8-s + (1.06 + 0.455i)9-s + (2.17 − 0.352i)10-s + (−3.85 + 1.64i)11-s + (2.35 + 1.23i)12-s + (1.14 − 3.41i)13-s + (6.59 − 3.46i)14-s + (0.947 − 1.16i)15-s + (3.43 + 2.17i)16-s + (−3.76 − 0.611i)17-s + ⋯ |
| L(s) = 1 | + (−1.12 + 0.845i)2-s + (−0.156 + 0.767i)3-s + (0.273 − 0.943i)4-s + (−0.437 − 0.229i)5-s + (−0.472 − 0.995i)6-s + (−1.39 − 0.226i)7-s + (−0.00865 − 0.0228i)8-s + (0.356 + 0.151i)9-s + (0.686 − 0.111i)10-s + (−1.16 + 0.495i)11-s + (0.681 + 0.357i)12-s + (0.318 − 0.947i)13-s + (1.76 − 0.925i)14-s + (0.244 − 0.299i)15-s + (0.859 + 0.543i)16-s + (−0.912 − 0.148i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0274770 - 0.0369324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0274770 - 0.0369324i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.14 + 3.41i)T \) |
| good | 2 | \( 1 + (1.59 - 1.19i)T + (0.556 - 1.92i)T^{2} \) |
| 3 | \( 1 + (0.271 - 1.32i)T + (-2.75 - 1.17i)T^{2} \) |
| 5 | \( 1 + (0.978 + 0.513i)T + (2.84 + 4.11i)T^{2} \) |
| 7 | \( 1 + (3.69 + 0.600i)T + (6.63 + 2.21i)T^{2} \) |
| 11 | \( 1 + (3.85 - 1.64i)T + (7.61 - 7.93i)T^{2} \) |
| 17 | \( 1 + (3.76 + 0.611i)T + (16.1 + 5.38i)T^{2} \) |
| 19 | \( 1 + (-1.11 + 1.93i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.105 + 0.182i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.73 - 2.80i)T + (8.06 - 27.8i)T^{2} \) |
| 31 | \( 1 + (2.91 - 4.22i)T + (-10.9 - 28.9i)T^{2} \) |
| 37 | \( 1 + (-7.40 - 0.597i)T + (36.5 + 5.93i)T^{2} \) |
| 41 | \( 1 + (2.04 - 9.99i)T + (-37.7 - 16.0i)T^{2} \) |
| 43 | \( 1 + (11.3 - 0.919i)T + (42.4 - 6.89i)T^{2} \) |
| 47 | \( 1 + (6.89 + 1.69i)T + (41.6 + 21.8i)T^{2} \) |
| 53 | \( 1 + (-0.891 - 2.35i)T + (-39.6 + 35.1i)T^{2} \) |
| 59 | \( 1 + (-1.75 + 1.11i)T + (25.2 - 53.3i)T^{2} \) |
| 61 | \( 1 + (4.20 + 5.15i)T + (-12.2 + 59.7i)T^{2} \) |
| 67 | \( 1 + (0.693 + 2.39i)T + (-56.6 + 35.8i)T^{2} \) |
| 71 | \( 1 + (-11.7 + 3.91i)T + (56.7 - 42.6i)T^{2} \) |
| 73 | \( 1 + (0.335 - 2.76i)T + (-70.8 - 17.4i)T^{2} \) |
| 79 | \( 1 + (-9.21 - 2.27i)T + (69.9 + 36.7i)T^{2} \) |
| 83 | \( 1 + (7.54 - 6.68i)T + (10.0 - 82.3i)T^{2} \) |
| 89 | \( 1 + (4.88 + 8.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0544 - 1.35i)T + (-96.6 - 7.80i)T^{2} \) |
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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
−13.15966464433498717767672341948, −12.79884157285441243541267198295, −10.98707927201116084941630323395, −10.03398780719756050508614344811, −9.604526621535894755638423500795, −8.356626811899581243282713873040, −7.41115349568772198562868678763, −6.40412453150343980848890076161, −4.94178998346688585651696742817, −3.39847021459443431285829726885,
0.05914514000256636842051170807, 2.12669063980175768401708134494, 3.58058018826777982163682906354, 5.91372201910447141624790949922, 7.05507638287884450584171794799, 8.075238915579832630363397120604, 9.252726992054445266348963047851, 9.990376752462550648014457540788, 11.11260346511931071121023407476, 11.84763458245171584667255292358
