| L(s) = 1 | + (−2.73 − 0.732i)2-s + (13.7 + 7.91i)3-s + (6.92 + 4i)4-s + (24.4 − 6.55i)5-s + (−31.6 − 31.6i)6-s + (35.2 − 61.1i)7-s + (−15.9 − 16i)8-s + (84.7 + 146. i)9-s − 71.6·10-s − 114. i·11-s + (63.3 + 109. i)12-s + (−251. + 67.2i)13-s + (−141. + 141. i)14-s + (387. + 103. i)15-s + (31.9 + 55.4i)16-s + (−54.2 + 202. i)17-s + ⋯ |
| L(s) = 1 | + (−0.683 − 0.183i)2-s + (1.52 + 0.879i)3-s + (0.433 + 0.250i)4-s + (0.978 − 0.262i)5-s + (−0.879 − 0.879i)6-s + (0.720 − 1.24i)7-s + (−0.249 − 0.250i)8-s + (1.04 + 1.81i)9-s − 0.716·10-s − 0.943i·11-s + (0.439 + 0.761i)12-s + (−1.48 + 0.398i)13-s + (−0.720 + 0.720i)14-s + (1.72 + 0.460i)15-s + (0.124 + 0.216i)16-s + (−0.187 + 0.700i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.29178 + 0.152224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29178 + 0.152224i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.73 + 0.732i)T \) |
| 37 | \( 1 + (-268. - 1.34e3i)T \) |
| good | 3 | \( 1 + (-13.7 - 7.91i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-24.4 + 6.55i)T + (541. - 312.5i)T^{2} \) |
| 7 | \( 1 + (-35.2 + 61.1i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 114. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + (251. - 67.2i)T + (2.47e4 - 1.42e4i)T^{2} \) |
| 17 | \( 1 + (54.2 - 202. i)T + (-7.23e4 - 4.17e4i)T^{2} \) |
| 19 | \( 1 + (-536. + 143. i)T + (1.12e5 - 6.51e4i)T^{2} \) |
| 23 | \( 1 + (105. + 105. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + (451. - 451. i)T - 7.07e5iT^{2} \) |
| 31 | \( 1 + (976. - 976. i)T - 9.23e5iT^{2} \) |
| 41 | \( 1 + (-82.9 - 47.8i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-845. - 845. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 3.37e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + (1.75e3 + 3.04e3i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-297. + 1.10e3i)T + (-1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (342. + 1.27e3i)T + (-1.19e7 + 6.92e6i)T^{2} \) |
| 67 | \( 1 + (-6.73e3 - 3.88e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + (-65.5 + 113. i)T + (-1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + 6.76e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (4.18e3 - 1.12e3i)T + (3.37e7 - 1.94e7i)T^{2} \) |
| 83 | \( 1 + (4.92e3 + 8.53e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-2.39e3 - 642. i)T + (5.43e7 + 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-4.89e3 - 4.89e3i)T + 8.85e7iT^{2} \) |
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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
−14.11319610210604809628741493272, −13.15877640668819049049716881278, −11.15199482264383577237734989590, −10.04245878515457826121391703717, −9.456273714303977571096031749709, −8.333292193228709034225651977073, −7.30493005343461603371101670623, −4.89590647288922043977074141818, −3.32093932303384594560797017920, −1.71953747762891293465443204212,
1.91862730339428891985811822064, 2.54947260773761458277847195338, 5.52151478655917188088666298647, 7.24179242782789886062519111342, 7.919367106114628343164700081636, 9.419413511714515809089509496068, 9.612257122201043070042426289254, 11.81590286213913647389693111184, 12.79565369236791609203950754141, 14.13336885263926406605356958593
