| 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
−13.87014792788713171210771684230, −13.00362143200220787080624515734, −11.58059672735387480369116486040, −10.59079775510388968229803525298, −9.290142150068854612827569602375, −8.319585888022682501043896997969, −7.15592431481289670004725768679, −5.44402105153650359361050414914, −2.99608137777603425061771200780, −1.64576368428541498213902261616,
1.79777286406446822387010207526, 4.21828626512886011728881254299, 5.79421094427130819279007673315, 7.30165791570601187833405077535, 8.747138448255011978517352453017, 9.461286812454275012790645185823, 10.41334985151441265680385685934, 11.94214211014225359536818128300, 13.46004256792878674703199976044, 14.54990644342933835130794333230
