L(s) = 1 | − 9.59i·2-s + (−8.65 − 12.9i)3-s − 60.0·4-s + 42.3·5-s + (−124. + 83.0i)6-s + 238. i·7-s + 269. i·8-s + (−93.1 + 224. i)9-s − 406. i·10-s − 261.·11-s + (519. + 778. i)12-s − 36.2·13-s + 2.29e3·14-s + (−366. − 548. i)15-s + 662.·16-s − 1.88e3·17-s + ⋯ |
L(s) = 1 | − 1.69i·2-s + (−0.555 − 0.831i)3-s − 1.87·4-s + 0.757·5-s + (−1.41 + 0.941i)6-s + 1.84i·7-s + 1.48i·8-s + (−0.383 + 0.923i)9-s − 1.28i·10-s − 0.651·11-s + (1.04 + 1.56i)12-s − 0.0594·13-s + 3.12·14-s + (−0.420 − 0.629i)15-s + 0.646·16-s − 1.57·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.365691 + 0.0595668i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.365691 + 0.0595668i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (8.65 + 12.9i)T \) |
| 23 | \( 1 + (-666. + 2.44e3i)T \) |
good | 2 | \( 1 + 9.59iT - 32T^{2} \) |
| 5 | \( 1 - 42.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 238. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 261.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 36.2T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.88e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.30e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 - 2.23e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 3.17e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.19e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.39e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.15e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 104. iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.04e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.22e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.96e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 5.31e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 7.48e3iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 3.49e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.16e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 4.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.50e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.61e4iT - 8.58e9T^{2} \) |
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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
−13.01799000887494359915908044507, −12.70372445499233936896590222591, −11.63387330381144580828659954414, −10.79379938158577748914818720787, −9.463748444419698230700240466944, −8.445330168858722075488390027782, −6.20014203253624962567525694475, −5.04276543785690119175256223112, −2.56041434805179888926254255085, −1.88970258970903522934286002050,
0.16983571518075808356543045718, 4.15759917083037697496215191508, 5.18785125461017519951500535889, 6.47635715461028449628343180484, 7.39950473954919807597074262311, 8.988675316528627822138161502721, 10.07957209261018346652812051916, 11.07804235894121378870867624078, 13.45004725319626138212481426344, 13.69961669080047612186303304949