L(s) = 1 | + (−3.62 − 3.62i)2-s + (4.19 + 3.07i)3-s + 18.2i·4-s + (10.8 + 10.8i)5-s + (−4.05 − 26.3i)6-s + (−5.52 − 5.52i)7-s + (37.1 − 37.1i)8-s + (8.11 + 25.7i)9-s − 78.9i·10-s + (25.6 − 25.6i)11-s + (−56.1 + 76.5i)12-s + (6.01 + 46.4i)13-s + 40.0i·14-s + (12.1 + 79.1i)15-s − 123.·16-s − 56.1·17-s + ⋯ |
L(s) = 1 | + (−1.28 − 1.28i)2-s + (0.806 + 0.591i)3-s + 2.28i·4-s + (0.974 + 0.974i)5-s + (−0.275 − 1.79i)6-s + (−0.298 − 0.298i)7-s + (1.64 − 1.64i)8-s + (0.300 + 0.953i)9-s − 2.49i·10-s + (0.702 − 0.702i)11-s + (−1.35 + 1.84i)12-s + (0.128 + 0.991i)13-s + 0.764i·14-s + (0.209 + 1.36i)15-s − 1.92·16-s − 0.800·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.982587 - 0.103547i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.982587 - 0.103547i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.19 - 3.07i)T \) |
| 13 | \( 1 + (-6.01 - 46.4i)T \) |
good | 2 | \( 1 + (3.62 + 3.62i)T + 8iT^{2} \) |
| 5 | \( 1 + (-10.8 - 10.8i)T + 125iT^{2} \) |
| 7 | \( 1 + (5.52 + 5.52i)T + 343iT^{2} \) |
| 11 | \( 1 + (-25.6 + 25.6i)T - 1.33e3iT^{2} \) |
| 17 | \( 1 + 56.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-72.4 + 72.4i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + 8.33T + 1.21e4T^{2} \) |
| 29 | \( 1 + 125. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (89.8 - 89.8i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (274. + 274. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + (44.2 + 44.2i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 - 100. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (-82.5 + 82.5i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 - 31.7iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-150. + 150. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 - 383.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-227. + 227. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + (-298. - 298. i)T + 3.57e5iT^{2} \) |
| 73 | \( 1 + (560. + 560. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 805.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (828. + 828. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (373. - 373. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 + (-254. + 254. i)T - 9.12e5iT^{2} \) |
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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
−16.04414624350346081066452450077, −14.21755729410932393412623974956, −13.40022171523488260208642911084, −11.45136131335261114619425289016, −10.55602639398446889189009705442, −9.543777624141278477849741206753, −8.828445818389145079762878157604, −7.00458701360794975172685312071, −3.57326074421142184281396547483, −2.17875027477075681236917098370,
1.43450353910668014596437465252, 5.60201490497414815892384077610, 6.90259487881726929124745821695, 8.280273112539328384576492702109, 9.191709054615361665758340211974, 9.943397485024478693078879844142, 12.53144407077574206095924254010, 13.75011438283667766450110856644, 14.88549647271682063349405311085, 15.91063375499830468259704920888