L(s) = 1 | + (1.12 − 1.94i)2-s + (0.277 − 0.480i)3-s + (−1.52 − 2.64i)4-s − 1.44·5-s + (−0.623 − 1.07i)6-s + (−1.02 − 1.77i)7-s − 2.35·8-s + (1.34 + 2.33i)9-s + (−1.62 + 2.81i)10-s + (1.27 − 2.21i)11-s − 1.69·12-s − 4.60·14-s + (−0.400 + 0.694i)15-s + (0.400 − 0.694i)16-s + (2.64 + 4.58i)17-s + 6.04·18-s + ⋯ |
L(s) = 1 | + (0.794 − 1.37i)2-s + (0.160 − 0.277i)3-s + (−0.762 − 1.32i)4-s − 0.646·5-s + (−0.254 − 0.440i)6-s + (−0.387 − 0.670i)7-s − 0.833·8-s + (0.448 + 0.777i)9-s + (−0.513 + 0.889i)10-s + (0.385 − 0.667i)11-s − 0.488·12-s − 1.23·14-s + (−0.103 + 0.179i)15-s + (0.100 − 0.173i)16-s + (0.642 + 1.11i)17-s + 1.42·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.702135 - 1.43030i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.702135 - 1.43030i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (-1.12 + 1.94i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.277 + 0.480i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 + (1.02 + 1.77i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.27 + 2.21i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.64 - 4.58i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.92 - 5.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.945 + 1.63i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.13 - 1.96i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.26T + 31T^{2} \) |
| 37 | \( 1 + (2.67 - 4.63i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.637 - 1.10i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.06 + 5.31i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2.95T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 + (-6.10 - 10.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.28 + 7.41i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.288 - 0.499i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.29 - 3.97i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 - 7.72T + 83T^{2} \) |
| 89 | \( 1 + (3.30 - 5.72i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.96 + 10.3i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.40906186397483256211144051925, −11.61041980441695972904060873771, −10.57893161583639180765487849826, −10.00325808879200294697827716316, −8.318256106869341408267950777152, −7.25731000500646181093224426441, −5.58180037649658538309998755809, −4.10040853589825648168981381951, −3.37016327596816426240421462671, −1.52689883213530529365582765022,
3.37132537688323264529287661625, 4.51962889787279237510927532994, 5.63052577988177386537391055901, 6.90642977775259276233257201906, 7.51668844787155801207694639329, 8.939379714636231942686446275645, 9.735891870165948494256853100657, 11.56442217790137039574434957359, 12.37052849792068554718385194965, 13.31293592486676508382710946243