| L(s) = 1 | + (1.91 − 0.563i)2-s + (1.96 + 2.26i)3-s + (3.36 − 2.16i)4-s + (1.91 + 13.3i)5-s + (5.04 + 3.24i)6-s + (−11.4 + 25.0i)7-s + (5.23 − 6.04i)8-s + (−1.28 + 8.90i)9-s + (11.1 + 24.4i)10-s + (−49.7 − 14.6i)11-s + (11.5 + 3.38i)12-s + (−8.02 − 17.5i)13-s + (−7.85 + 54.6i)14-s + (−26.4 + 30.4i)15-s + (6.64 − 14.5i)16-s + (99.0 + 63.6i)17-s + ⋯ |
| L(s) = 1 | + (0.678 − 0.199i)2-s + (0.378 + 0.436i)3-s + (0.420 − 0.270i)4-s + (0.171 + 1.19i)5-s + (0.343 + 0.220i)6-s + (−0.618 + 1.35i)7-s + (0.231 − 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.353 + 0.773i)10-s + (−1.36 − 0.400i)11-s + (0.276 + 0.0813i)12-s + (−0.171 − 0.374i)13-s + (−0.149 + 1.04i)14-s + (−0.454 + 0.524i)15-s + (0.103 − 0.227i)16-s + (1.41 + 0.908i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.89714 + 1.58722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89714 + 1.58722i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.91 + 0.563i)T \) |
| 3 | \( 1 + (-1.96 - 2.26i)T \) |
| 23 | \( 1 + (-60.7 + 92.0i)T \) |
| good | 5 | \( 1 + (-1.91 - 13.3i)T + (-119. + 35.2i)T^{2} \) |
| 7 | \( 1 + (11.4 - 25.0i)T + (-224. - 259. i)T^{2} \) |
| 11 | \( 1 + (49.7 + 14.6i)T + (1.11e3 + 719. i)T^{2} \) |
| 13 | \( 1 + (8.02 + 17.5i)T + (-1.43e3 + 1.66e3i)T^{2} \) |
| 17 | \( 1 + (-99.0 - 63.6i)T + (2.04e3 + 4.46e3i)T^{2} \) |
| 19 | \( 1 + (-75.4 + 48.4i)T + (2.84e3 - 6.23e3i)T^{2} \) |
| 29 | \( 1 + (-188. - 121. i)T + (1.01e4 + 2.21e4i)T^{2} \) |
| 31 | \( 1 + (42.6 - 49.2i)T + (-4.23e3 - 2.94e4i)T^{2} \) |
| 37 | \( 1 + (7.10 - 49.4i)T + (-4.86e4 - 1.42e4i)T^{2} \) |
| 41 | \( 1 + (64.9 + 451. i)T + (-6.61e4 + 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-269. - 311. i)T + (-1.13e4 + 7.86e4i)T^{2} \) |
| 47 | \( 1 - 347.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-52.4 + 114. i)T + (-9.74e4 - 1.12e5i)T^{2} \) |
| 59 | \( 1 + (266. + 583. i)T + (-1.34e5 + 1.55e5i)T^{2} \) |
| 61 | \( 1 + (383. - 442. i)T + (-3.23e4 - 2.24e5i)T^{2} \) |
| 67 | \( 1 + (735. - 215. i)T + (2.53e5 - 1.62e5i)T^{2} \) |
| 71 | \( 1 + (412. - 121. i)T + (3.01e5 - 1.93e5i)T^{2} \) |
| 73 | \( 1 + (293. - 188. i)T + (1.61e5 - 3.53e5i)T^{2} \) |
| 79 | \( 1 + (55.2 + 120. i)T + (-3.22e5 + 3.72e5i)T^{2} \) |
| 83 | \( 1 + (-211. + 1.47e3i)T + (-5.48e5 - 1.61e5i)T^{2} \) |
| 89 | \( 1 + (-296. - 342. i)T + (-1.00e5 + 6.97e5i)T^{2} \) |
| 97 | \( 1 + (-55.0 - 383. i)T + (-8.75e5 + 2.57e5i)T^{2} \) |
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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
−12.90579074982619737813174910173, −12.10448155503296931152010739317, −10.67746687200441333960705306080, −10.23566177081321687585778753116, −8.847888771462885799574713878213, −7.49382932979979791339596732025, −6.07380866972646227504327889687, −5.21723380431846176279279935597, −3.09952340446527564221267955684, −2.74900336395430578846745892941,
1.01094110508667210427273772220, 3.08194755535428172869229057440, 4.55290459997610297408824930495, 5.64573799319855867217086795946, 7.30642997839229794070770243265, 7.80686395154144957754972714395, 9.421217038376366339796649738815, 10.31953453818256370347491909742, 11.95130621390141519979601626385, 12.70878761929087816847560489995
