| L(s) = 1 | + (−3.75 − 1.36i)2-s + (−23.1 + 8.41i)3-s + (12.2 + 10.2i)4-s + (6.03 − 34.2i)5-s + 98.3·6-s + (19.4 − 110. i)7-s + (−32.0 − 55.4i)8-s + (277. − 232. i)9-s + (−69.5 + 120. i)10-s + (−141. − 244. i)11-s + (−369. − 134. i)12-s + (186. + 156. i)13-s + (−223. + 387. i)14-s + (148. + 842. i)15-s + (44.4 + 252. i)16-s + (−470. + 395. i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (−1.48 + 0.539i)3-s + (0.383 + 0.321i)4-s + (0.107 − 0.612i)5-s + 1.11·6-s + (0.149 − 0.850i)7-s + (−0.176 − 0.306i)8-s + (1.14 − 0.957i)9-s + (−0.219 + 0.380i)10-s + (−0.351 − 0.608i)11-s + (−0.741 − 0.269i)12-s + (0.305 + 0.256i)13-s + (−0.305 + 0.528i)14-s + (0.170 + 0.966i)15-s + (0.0434 + 0.246i)16-s + (−0.395 + 0.331i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.174254 + 0.219106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174254 + 0.219106i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.75 + 1.36i)T \) |
| 37 | \( 1 + (4.20e3 - 7.18e3i)T \) |
| good | 3 | \( 1 + (23.1 - 8.41i)T + (186. - 156. i)T^{2} \) |
| 5 | \( 1 + (-6.03 + 34.2i)T + (-2.93e3 - 1.06e3i)T^{2} \) |
| 7 | \( 1 + (-19.4 + 110. i)T + (-1.57e4 - 5.74e3i)T^{2} \) |
| 11 | \( 1 + (141. + 244. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-186. - 156. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (470. - 395. i)T + (2.46e5 - 1.39e6i)T^{2} \) |
| 19 | \( 1 + (876. - 318. i)T + (1.89e6 - 1.59e6i)T^{2} \) |
| 23 | \( 1 + (1.68e3 - 2.92e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (397. + 688. i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + 886.T + 2.86e7T^{2} \) |
| 41 | \( 1 + (2.58e3 + 2.16e3i)T + (2.01e7 + 1.14e8i)T^{2} \) |
| 43 | \( 1 + 6.00e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (7.82e3 - 1.35e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-2.84e3 - 1.61e4i)T + (-3.92e8 + 1.43e8i)T^{2} \) |
| 59 | \( 1 + (4.26e3 + 2.41e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (-3.42e4 - 2.87e4i)T + (1.46e8 + 8.31e8i)T^{2} \) |
| 67 | \( 1 + (393. - 2.23e3i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (-1.06e4 + 3.86e3i)T + (1.38e9 - 1.15e9i)T^{2} \) |
| 73 | \( 1 - 8.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (1.36e4 - 7.74e4i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (9.39e4 - 7.88e4i)T + (6.84e8 - 3.87e9i)T^{2} \) |
| 89 | \( 1 + (-6.86e3 - 3.89e4i)T + (-5.24e9 + 1.90e9i)T^{2} \) |
| 97 | \( 1 + (9.18e4 - 1.59e5i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−13.64525218781794529986113410808, −12.47383897761936545707799102059, −11.32891892113311681297426106634, −10.71106578371112390793650042657, −9.693939644416805547504227457558, −8.269044850494817146854461219687, −6.68215554497417173794717585550, −5.40800350603962863532183941106, −4.03431266842568544757494846418, −1.15492013140647387055416725212,
0.21111973328630408740311427778, 2.15072997506832518314910719929, 5.09456329168659255630092603250, 6.24949069205462554685462559631, 7.06494088917531987646928895294, 8.562628104456208645024848425355, 10.19062184691352949574658731707, 11.00628287085232759283431221938, 11.98731251447170016284576133090, 12.90323987064318162493149266241
