L(s) = 1 | + (−2.10 + 1.21i)2-s + (3.73 − 6.46i)3-s + (−5.05 + 8.75i)4-s + (11.6 − 22.1i)5-s + 18.1i·6-s + (22.9 − 43.3i)7-s − 63.3i·8-s + (12.6 + 21.8i)9-s + (2.35 + 60.6i)10-s + (95.0 − 164. i)11-s + (37.7 + 65.3i)12-s + 63.2·13-s + (4.42 + 118. i)14-s + (−99.5 − 157. i)15-s + (−3.93 − 6.81i)16-s + (−166. + 288. i)17-s + ⋯ |
L(s) = 1 | + (−0.525 + 0.303i)2-s + (0.414 − 0.718i)3-s + (−0.315 + 0.547i)4-s + (0.466 − 0.884i)5-s + 0.503i·6-s + (0.467 − 0.884i)7-s − 0.990i·8-s + (0.155 + 0.269i)9-s + (0.0235 + 0.606i)10-s + (0.785 − 1.36i)11-s + (0.262 + 0.453i)12-s + 0.374·13-s + (0.0225 + 0.606i)14-s + (−0.442 − 0.701i)15-s + (−0.0153 − 0.0266i)16-s + (−0.575 + 0.996i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.15641 - 0.519527i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15641 - 0.519527i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-11.6 + 22.1i)T \) |
| 7 | \( 1 + (-22.9 + 43.3i)T \) |
good | 2 | \( 1 + (2.10 - 1.21i)T + (8 - 13.8i)T^{2} \) |
| 3 | \( 1 + (-3.73 + 6.46i)T + (-40.5 - 70.1i)T^{2} \) |
| 11 | \( 1 + (-95.0 + 164. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 - 63.2T + 2.85e4T^{2} \) |
| 17 | \( 1 + (166. - 288. i)T + (-4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (343. - 198. i)T + (6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (361. - 208. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 - 1.38e3T + 7.07e5T^{2} \) |
| 31 | \( 1 + (-416. - 240. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (1.36e3 - 788. i)T + (9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 - 959. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.68e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (-868. - 1.50e3i)T + (-2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-2.06e3 - 1.19e3i)T + (3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (-131. - 76.0i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (2.35e3 - 1.35e3i)T + (6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-2.68e3 - 1.55e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 739.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (2.97e3 - 5.14e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-1.71e3 - 2.97e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 - 1.09e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + (2.30e3 - 1.32e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + 5.79e3T + 8.85e7T^{2} \) |
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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
−16.09316636374426924533042161892, −13.92392785426157350011241864739, −13.45338858380829925206629189127, −12.26229018820420607009256442629, −10.41161229037126117570235425245, −8.605726874918311116770106784981, −8.197424160893559208416891909850, −6.50998590669358350362979104618, −4.13419050154590524545291388731, −1.20443045161012741216697131796,
2.23059667538637953409462863522, 4.63651764185132023479331374934, 6.54067366549527496996977554374, 8.788433576970449811671959604984, 9.624655657930705674949582588673, 10.59620483461028847626822602166, 11.97975044876825466935341961098, 13.99590439339910975879586079220, 14.85092536823310166402760803188, 15.51823856908033037109892492188