L(s) = 1 | + (1.20 − 1.20i)2-s + 5.09i·4-s + (−2.60 + 6.29i)5-s + (−5.31 − 12.8i)7-s + (15.7 + 15.7i)8-s + (4.44 + 10.7i)10-s + (−28.4 + 11.8i)11-s + 66.0i·13-s + (−21.8 − 9.05i)14-s − 2.65·16-s + (−3.91 + 69.9i)17-s + (−56.3 + 56.3i)19-s + (−32.0 − 13.2i)20-s + (−20.1 + 48.5i)22-s + (26.6 − 11.0i)23-s + ⋯ |
L(s) = 1 | + (0.426 − 0.426i)2-s + 0.636i·4-s + (−0.233 + 0.562i)5-s + (−0.286 − 0.692i)7-s + (0.697 + 0.697i)8-s + (0.140 + 0.339i)10-s + (−0.780 + 0.323i)11-s + 1.40i·13-s + (−0.417 − 0.172i)14-s − 0.0414·16-s + (−0.0558 + 0.998i)17-s + (−0.680 + 0.680i)19-s + (−0.358 − 0.148i)20-s + (−0.195 + 0.470i)22-s + (0.242 − 0.100i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00939 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.00939 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.08580 + 1.07564i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08580 + 1.07564i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 + (3.91 - 69.9i)T \) |
good | 2 | \( 1 + (-1.20 + 1.20i)T - 8iT^{2} \) |
| 5 | \( 1 + (2.60 - 6.29i)T + (-88.3 - 88.3i)T^{2} \) |
| 7 | \( 1 + (5.31 + 12.8i)T + (-242. + 242. i)T^{2} \) |
| 11 | \( 1 + (28.4 - 11.8i)T + (941. - 941. i)T^{2} \) |
| 13 | \( 1 - 66.0iT - 2.19e3T^{2} \) |
| 19 | \( 1 + (56.3 - 56.3i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + (-26.6 + 11.0i)T + (8.60e3 - 8.60e3i)T^{2} \) |
| 29 | \( 1 + (-101. + 245. i)T + (-1.72e4 - 1.72e4i)T^{2} \) |
| 31 | \( 1 + (33.0 + 13.6i)T + (2.10e4 + 2.10e4i)T^{2} \) |
| 37 | \( 1 + (-330. - 136. i)T + (3.58e4 + 3.58e4i)T^{2} \) |
| 41 | \( 1 + (12.5 + 30.2i)T + (-4.87e4 + 4.87e4i)T^{2} \) |
| 43 | \( 1 + (364. + 364. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 - 210. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (61.6 - 61.6i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (219. + 219. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (-139. - 337. i)T + (-1.60e5 + 1.60e5i)T^{2} \) |
| 67 | \( 1 - 660.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-367. - 152. i)T + (2.53e5 + 2.53e5i)T^{2} \) |
| 73 | \( 1 + (-246. + 594. i)T + (-2.75e5 - 2.75e5i)T^{2} \) |
| 79 | \( 1 + (355. - 147. i)T + (3.48e5 - 3.48e5i)T^{2} \) |
| 83 | \( 1 + (108. - 108. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 + 599. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (17.0 - 41.1i)T + (-6.45e5 - 6.45e5i)T^{2} \) |
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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
−12.80940303696967243769165111336, −11.77491098162558087585642890411, −10.90831838874235988184830231845, −10.00199333881536584167638460712, −8.457136396953635365795828392286, −7.47220133301924017977454449715, −6.44141019187309158729408022575, −4.54143707271566186781558261297, −3.65296714689534970014997696915, −2.19611943799874642721036130634,
0.63217503814916052706373984990, 2.85417761610707792527095892233, 4.81234742656066470481589027836, 5.48050949881910264918134953960, 6.69311563297052656500091386037, 8.034267145417602310852181501772, 9.122021290649502832344744814239, 10.25926152210607200072842358103, 11.19706678880279247250948019803, 12.69172136245727300719137458801