L(s) = 1 | + (−0.402 + 1.23i)2-s + (−1.05 + 0.765i)3-s + (0.244 + 0.177i)4-s + (−1.11 − 3.42i)5-s + (−0.524 − 1.61i)6-s + (0.809 + 0.587i)7-s + (−2.42 + 1.76i)8-s + (−0.402 + 1.23i)9-s + 4.69·10-s − 0.394·12-s + (−0.187 + 0.575i)13-s + (−1.05 + 0.765i)14-s + (3.80 + 2.76i)15-s + (−1.02 − 3.14i)16-s + (1.94 + 5.99i)17-s + (−1.37 − 0.997i)18-s + ⋯ |
L(s) = 1 | + (−0.284 + 0.876i)2-s + (−0.608 + 0.442i)3-s + (0.122 + 0.0889i)4-s + (−0.498 − 1.53i)5-s + (−0.214 − 0.658i)6-s + (0.305 + 0.222i)7-s + (−0.858 + 0.623i)8-s + (−0.134 + 0.413i)9-s + 1.48·10-s − 0.113·12-s + (−0.0518 + 0.159i)13-s + (−0.281 + 0.204i)14-s + (0.981 + 0.712i)15-s + (−0.255 − 0.785i)16-s + (0.472 + 1.45i)17-s + (−0.323 − 0.235i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0954400 - 0.192514i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0954400 - 0.192514i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.402 - 1.23i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (1.05 - 0.765i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (1.11 + 3.42i)T + (-4.04 + 2.93i)T^{2} \) |
| 13 | \( 1 + (0.187 - 0.575i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.94 - 5.99i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.42 + 1.76i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 6.30T + 23T^{2} \) |
| 29 | \( 1 + (6.71 + 4.88i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (7.45 + 5.41i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (5.66 - 4.11i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 5.30T + 43T^{2} \) |
| 47 | \( 1 + (7.20 - 5.23i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.646 - 1.98i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (8.33 + 6.05i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.215 + 0.663i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 9.51T + 67T^{2} \) |
| 71 | \( 1 + (0.215 + 0.663i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.04 - 2.93i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (1.45 - 4.46i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.927 - 2.85i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 + (-1.11 + 3.42i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.80507962240783351088668585212, −9.649139314252215677628286910486, −8.830013844823603357831340903031, −7.995895791002138686567369463834, −7.73391687316774827326648603883, −6.17568012573830696070619238070, −5.56396556210601190251746999365, −4.78843855750364345036820536173, −3.78585468440581452774022362349, −1.88230470545999933646424423264,
0.11804911461484907480070043421, 1.69826877095568580312838663160, 3.05297974978714713277355616188, 3.59789004871286936346042687299, 5.37582841154722148216846525924, 6.33223326813622793734828192988, 7.06129588733101442911483812466, 7.64158468417295235110162228840, 9.088596468987496533303374984587, 10.05401390980560964118400827216