L(s) = 1 | − 4·2-s + (1.84 + 15.4i)3-s + 16·4-s + 87.9i·5-s + (−7.38 − 61.9i)6-s − 193.·7-s − 64·8-s + (−236. + 57.1i)9-s − 351. i·10-s − 206.·11-s + (29.5 + 247. i)12-s + 754. i·13-s + 774.·14-s + (−1.36e3 + 162. i)15-s + 256·16-s + 1.40e3i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.118 + 0.992i)3-s + 0.5·4-s + 1.57i·5-s + (−0.0837 − 0.702i)6-s − 1.49·7-s − 0.353·8-s + (−0.971 + 0.235i)9-s − 1.11i·10-s − 0.514·11-s + (0.0591 + 0.496i)12-s + 1.23i·13-s + 1.05·14-s + (−1.56 + 0.186i)15-s + 0.250·16-s + 1.17i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.5330975262\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5330975262\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + (-1.84 - 15.4i)T \) |
| 59 | \( 1 + (2.40e4 - 1.17e4i)T \) |
good | 5 | \( 1 - 87.9iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 193.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 206.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 754. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.40e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.39e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 859.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.01e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 6.20e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.37e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 2.92e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.46e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 8.21e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.66e4iT - 4.18e8T^{2} \) |
| 61 | \( 1 - 3.93e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 4.29e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.91e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 2.75e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 5.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.35e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.42e4iT - 8.58e9T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.92830153998069848097832274457, −10.39426906453489507989000979739, −9.758366024407279089598784823988, −8.920479551216981530383814977219, −7.71034389879258181790980939268, −6.46673919305462676643519528072, −6.15526280099339329612989459784, −4.13237361005277376841174209373, −3.18265874027775758872139905703, −2.32170003594354428717858432058,
0.30004762891708371389426482545, 0.56541709580024765887402939071, 2.15235489219549060750640987061, 3.34292995073492446940803861945, 5.19773481992303156964165105437, 6.08676286101109866673360969805, 7.17949126914406907051026808179, 8.100805197225059092315732996279, 8.872559443641026259587780401959, 9.585064466282881365000738001721