| L(s) = 1 | + (0.0559 − 0.639i)2-s + (0.432 − 1.67i)3-s + (1.56 + 0.275i)4-s + (2.15 + 0.602i)5-s + (−1.04 − 0.370i)6-s + (−2.47 + 3.52i)7-s + (0.595 − 2.22i)8-s + (−2.62 − 1.45i)9-s + (0.505 − 1.34i)10-s + (−0.157 + 0.433i)11-s + (1.13 − 2.50i)12-s + (−5.28 + 0.462i)13-s + (2.11 + 1.77i)14-s + (1.94 − 3.35i)15-s + (1.59 + 0.580i)16-s + (−1.32 − 4.93i)17-s + ⋯ |
| L(s) = 1 | + (0.0395 − 0.452i)2-s + (0.249 − 0.968i)3-s + (0.782 + 0.137i)4-s + (0.963 + 0.269i)5-s + (−0.427 − 0.151i)6-s + (−0.933 + 1.33i)7-s + (0.210 − 0.786i)8-s + (−0.875 − 0.483i)9-s + (0.159 − 0.424i)10-s + (−0.0476 + 0.130i)11-s + (0.328 − 0.722i)12-s + (−1.46 + 0.128i)13-s + (0.565 + 0.474i)14-s + (0.501 − 0.865i)15-s + (0.399 + 0.145i)16-s + (−0.320 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23821 - 0.630451i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23821 - 0.630451i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.432 + 1.67i)T \) |
| 5 | \( 1 + (-2.15 - 0.602i)T \) |
| good | 2 | \( 1 + (-0.0559 + 0.639i)T + (-1.96 - 0.347i)T^{2} \) |
| 7 | \( 1 + (2.47 - 3.52i)T + (-2.39 - 6.57i)T^{2} \) |
| 11 | \( 1 + (0.157 - 0.433i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (5.28 - 0.462i)T + (12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (1.32 + 4.93i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.03 - 0.598i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.25 - 1.57i)T + (7.86 - 21.6i)T^{2} \) |
| 29 | \( 1 + (-0.00462 + 0.00387i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.387 - 2.19i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-7.25 + 1.94i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.02 + 3.60i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.72 + 0.806i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (-9.24 - 6.47i)T + (16.0 + 44.1i)T^{2} \) |
| 53 | \( 1 + (3.94 + 3.94i)T + 53iT^{2} \) |
| 59 | \( 1 + (-0.388 + 0.141i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.399 - 2.26i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.511 + 5.84i)T + (-65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.648 - 0.374i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-11.1 - 2.97i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.01 - 1.20i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (10.6 + 0.930i)T + (81.7 + 14.4i)T^{2} \) |
| 89 | \( 1 + (6.36 + 11.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.62 + 7.76i)T + (-62.3 + 74.3i)T^{2} \) |
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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
−12.68043059165394165459266100256, −12.35985186806153252201025708119, −11.32445501467387748792339117273, −9.828796705119301075190050869904, −9.164276509123550634879051179259, −7.45732668427877994193008613423, −6.56956165840745198225085278911, −5.63022950190780816228929301320, −2.78845143431603414451020655729, −2.27552976879847012931635473009,
2.58367213878971210092623084320, 4.30551436460860189816696069141, 5.72054225773121973034384315490, 6.74431935559413573330179156822, 8.011626443612662151488306008166, 9.536822707123220494305484110342, 10.22335603049437113457332678342, 10.90848725145821161509102984701, 12.53872677728332952187089509995, 13.63334131145219814510757930820
