| L(s) = 1 | + (0.596 + 0.0240i)2-s + (0.703 − 2.43i)3-s + (−1.63 − 0.132i)4-s + (3.70 + 1.40i)5-s + (0.478 − 1.43i)6-s + (−0.324 − 0.762i)7-s + (−2.16 − 0.262i)8-s + (−2.87 − 1.81i)9-s + (2.17 + 0.926i)10-s + (−0.896 − 1.41i)11-s + (−1.47 + 3.88i)12-s + (−2.52 + 2.57i)13-s + (−0.175 − 0.462i)14-s + (6.01 − 8.00i)15-s + (1.96 + 0.318i)16-s + (6.43 − 2.74i)17-s + ⋯ |
| L(s) = 1 | + (0.422 + 0.0170i)2-s + (0.406 − 1.40i)3-s + (−0.818 − 0.0661i)4-s + (1.65 + 0.627i)5-s + (0.195 − 0.585i)6-s + (−0.122 − 0.288i)7-s + (−0.763 − 0.0927i)8-s + (−0.958 − 0.605i)9-s + (0.687 + 0.292i)10-s + (−0.270 − 0.427i)11-s + (−0.425 + 1.12i)12-s + (−0.699 + 0.714i)13-s + (−0.0469 − 0.123i)14-s + (1.55 − 2.06i)15-s + (0.490 + 0.0796i)16-s + (1.56 − 0.665i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.35928 - 0.752856i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35928 - 0.752856i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.52 - 2.57i)T \) |
| good | 2 | \( 1 + (-0.596 - 0.0240i)T + (1.99 + 0.160i)T^{2} \) |
| 3 | \( 1 + (-0.703 + 2.43i)T + (-2.53 - 1.60i)T^{2} \) |
| 5 | \( 1 + (-3.70 - 1.40i)T + (3.74 + 3.31i)T^{2} \) |
| 7 | \( 1 + (0.324 + 0.762i)T + (-4.84 + 5.04i)T^{2} \) |
| 11 | \( 1 + (0.896 + 1.41i)T + (-4.71 + 9.93i)T^{2} \) |
| 17 | \( 1 + (-6.43 + 2.74i)T + (11.7 - 12.2i)T^{2} \) |
| 19 | \( 1 + (6.17 - 3.56i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.23 - 2.13i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.333 - 8.28i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (-0.422 - 0.476i)T + (-3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (4.24 - 0.865i)T + (34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (3.37 + 0.977i)T + (34.6 + 21.9i)T^{2} \) |
| 43 | \( 1 + (-0.218 + 1.07i)T + (-39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (1.62 - 1.11i)T + (16.6 - 43.9i)T^{2} \) |
| 53 | \( 1 + (0.146 - 1.21i)T + (-51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (1.82 + 11.2i)T + (-55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (1.57 - 1.18i)T + (16.9 - 58.5i)T^{2} \) |
| 67 | \( 1 + (0.248 + 3.07i)T + (-66.1 + 10.7i)T^{2} \) |
| 71 | \( 1 + (2.34 - 2.24i)T + (2.85 - 70.9i)T^{2} \) |
| 73 | \( 1 + (-4.05 + 7.71i)T + (-41.4 - 60.0i)T^{2} \) |
| 79 | \( 1 + (3.06 + 4.44i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (1.84 - 7.49i)T + (-73.4 - 38.5i)T^{2} \) |
| 89 | \( 1 + (14.6 + 8.44i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0230 + 0.0188i)T + (19.4 + 95.0i)T^{2} \) |
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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
−12.85763581531037432616539492165, −12.19707425058941366029858197294, −10.41509626862670720307827880588, −9.603431327959235573532882140094, −8.498968384885938443360378988380, −7.18781998446995535484314650179, −6.24357055998728091756393597880, −5.28452575154974425133305177007, −3.16778407750757940894655670948, −1.74628867828756993236492666591,
2.67282562755406300532057182839, 4.26945292368217674609900172383, 5.17526434169166277470411324840, 5.93080903854298529010410017269, 8.320537498078993781348396825121, 9.144921965515437066481701437403, 9.998588179703306840586682935554, 10.26657428696973952293999719403, 12.38844250300988720927870545413, 13.01014485169274814415911187934
