L(s) = 1 | + (−1.53 + 1.28i)2-s + (3.31 + 2.77i)3-s + (0.694 − 3.93i)4-s + (10.2 − 3.72i)5-s − 8.64·6-s + (12.8 − 4.67i)7-s + (4.00 + 6.92i)8-s + (−1.44 − 8.17i)9-s + (−10.8 + 18.8i)10-s + (16.5 + 28.6i)11-s + (13.2 − 11.1i)12-s + (−4.61 + 26.1i)13-s + (−13.6 + 23.6i)14-s + (44.1 + 16.0i)15-s + (−15.0 − 5.47i)16-s + (8.03 + 45.5i)17-s + ⋯ |
L(s) = 1 | + (−0.541 + 0.454i)2-s + (0.637 + 0.534i)3-s + (0.0868 − 0.492i)4-s + (0.914 − 0.332i)5-s − 0.588·6-s + (0.693 − 0.252i)7-s + (0.176 + 0.306i)8-s + (−0.0533 − 0.302i)9-s + (−0.343 + 0.595i)10-s + (0.453 + 0.785i)11-s + (0.318 − 0.267i)12-s + (−0.0985 + 0.558i)13-s + (−0.260 + 0.451i)14-s + (0.760 + 0.276i)15-s + (−0.234 − 0.0855i)16-s + (0.114 + 0.649i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.53735 + 0.601697i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.53735 + 0.601697i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.53 - 1.28i)T \) |
| 37 | \( 1 + (133. + 180. i)T \) |
good | 3 | \( 1 + (-3.31 - 2.77i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (-10.2 + 3.72i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (-12.8 + 4.67i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (-16.5 - 28.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (4.61 - 26.1i)T + (-2.06e3 - 751. i)T^{2} \) |
| 17 | \( 1 + (-8.03 - 45.5i)T + (-4.61e3 + 1.68e3i)T^{2} \) |
| 19 | \( 1 + (-19.8 - 16.6i)T + (1.19e3 + 6.75e3i)T^{2} \) |
| 23 | \( 1 + (-21.3 + 37.0i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (87.7 + 152. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 191.T + 2.97e4T^{2} \) |
| 41 | \( 1 + (-1.84 + 10.4i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + 42.6T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-167. + 290. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-351. - 127. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-240. - 87.5i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (107. - 607. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (251. - 91.5i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (753. + 632. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-20.5 + 7.46i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (150. + 856. i)T + (-5.37e5 + 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-599. - 218. i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (661. - 1.14e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−14.54990644342933835130794333230, −13.46004256792878674703199976044, −11.94214211014225359536818128300, −10.41334985151441265680385685934, −9.461286812454275012790645185823, −8.747138448255011978517352453017, −7.30165791570601187833405077535, −5.79421094427130819279007673315, −4.21828626512886011728881254299, −1.79777286406446822387010207526,
1.64576368428541498213902261616, 2.99608137777603425061771200780, 5.44402105153650359361050414914, 7.15592431481289670004725768679, 8.319585888022682501043896997969, 9.290142150068854612827569602375, 10.59079775510388968229803525298, 11.58059672735387480369116486040, 13.00362143200220787080624515734, 13.87014792788713171210771684230