| L(s) = 1 | + (0.980 − 0.200i)2-s + (−2.27 − 0.370i)3-s + (−0.918 + 0.391i)4-s + (0.651 + 2.64i)5-s + (−2.30 + 0.0930i)6-s + (4.17 + 1.98i)7-s + (−2.46 + 1.70i)8-s + (2.20 + 0.736i)9-s + (1.16 + 2.46i)10-s + (0.585 + 1.75i)11-s + (2.23 − 0.551i)12-s + (−2.08 − 2.94i)13-s + (4.49 + 1.10i)14-s + (−0.505 − 6.25i)15-s + (−0.697 + 0.726i)16-s + (−2.65 + 5.60i)17-s + ⋯ |
| L(s) = 1 | + (0.693 − 0.141i)2-s + (−1.31 − 0.213i)3-s + (−0.459 + 0.195i)4-s + (0.291 + 1.18i)5-s + (−0.942 + 0.0379i)6-s + (1.57 + 0.749i)7-s + (−0.873 + 0.602i)8-s + (0.734 + 0.245i)9-s + (0.369 + 0.777i)10-s + (0.176 + 0.528i)11-s + (0.645 − 0.159i)12-s + (−0.577 − 0.816i)13-s + (1.20 + 0.296i)14-s + (−0.130 − 1.61i)15-s + (−0.174 + 0.181i)16-s + (−0.645 + 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.833242 + 0.566906i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.833242 + 0.566906i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.08 + 2.94i)T \) |
| good | 2 | \( 1 + (-0.980 + 0.200i)T + (1.83 - 0.783i)T^{2} \) |
| 3 | \( 1 + (2.27 + 0.370i)T + (2.84 + 0.950i)T^{2} \) |
| 5 | \( 1 + (-0.651 - 2.64i)T + (-4.42 + 2.32i)T^{2} \) |
| 7 | \( 1 + (-4.17 - 1.98i)T + (4.42 + 5.42i)T^{2} \) |
| 11 | \( 1 + (-0.585 - 1.75i)T + (-8.79 + 6.60i)T^{2} \) |
| 17 | \( 1 + (2.65 - 5.60i)T + (-10.7 - 13.1i)T^{2} \) |
| 19 | \( 1 + (-0.560 + 0.323i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.07 + 3.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.335 + 1.64i)T + (-26.6 + 11.3i)T^{2} \) |
| 31 | \( 1 + (-0.757 + 1.44i)T + (-17.6 - 25.5i)T^{2} \) |
| 37 | \( 1 + (2.53 + 4.01i)T + (-15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-0.803 + 4.94i)T + (-38.8 - 12.9i)T^{2} \) |
| 43 | \( 1 + (-7.23 - 4.57i)T + (18.4 + 38.8i)T^{2} \) |
| 47 | \( 1 + (-10.4 + 1.26i)T + (45.6 - 11.2i)T^{2} \) |
| 53 | \( 1 + (-1.29 - 1.86i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (6.37 - 6.12i)T + (2.37 - 58.9i)T^{2} \) |
| 61 | \( 1 + (7.16 + 0.578i)T + (60.2 + 9.78i)T^{2} \) |
| 67 | \( 1 + (-3.81 + 8.95i)T + (-46.4 - 48.3i)T^{2} \) |
| 71 | \( 1 + (-7.93 - 6.47i)T + (14.2 + 69.5i)T^{2} \) |
| 73 | \( 1 + (-0.757 + 0.854i)T + (-8.79 - 72.4i)T^{2} \) |
| 79 | \( 1 + (1.56 + 12.8i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (8.62 + 3.27i)T + (62.1 + 55.0i)T^{2} \) |
| 89 | \( 1 + (-7.26 - 4.19i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (10.9 - 3.15i)T + (81.9 - 51.8i)T^{2} \) |
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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
−12.60052101082478168649459090785, −12.13172498841473651617399244202, −11.06426971424199052287421261325, −10.57400522447577772410873372752, −8.859621731342472782314112684236, −7.60655799791035106844615982153, −6.20136659475424529622167058664, −5.40976325412232163823700651072, −4.41414734016117159781956739406, −2.43008409860835093119220172496,
0.988917781255826247250863003761, 4.41020125291715573191590609292, 4.90139743705021609355966300527, 5.58928417947527375516460298661, 7.06275656387659424177026185572, 8.646739727726439198037489316422, 9.577387253207613302528417726052, 10.96775757864365908418530845596, 11.66958639108844787984961266394, 12.49425209584842957404345659473
