L(s) = 1 | + (−1.35 + 0.690i)2-s + (−0.787 − 0.124i)3-s + (0.182 − 0.251i)4-s + (0.834 + 2.07i)5-s + (1.15 − 0.374i)6-s + (−0.0299 − 0.189i)7-s + (0.401 − 2.53i)8-s + (−2.24 − 0.730i)9-s + (−2.56 − 2.23i)10-s + (−0.175 + 0.175i)12-s + (−1.40 − 2.75i)13-s + (0.171 + 0.235i)14-s + (−0.398 − 1.73i)15-s + (1.39 + 4.30i)16-s + (1.68 − 3.30i)17-s + (3.54 − 0.562i)18-s + ⋯ |
L(s) = 1 | + (−0.957 + 0.487i)2-s + (−0.454 − 0.0720i)3-s + (0.0912 − 0.125i)4-s + (0.373 + 0.927i)5-s + (0.470 − 0.152i)6-s + (−0.0113 − 0.0714i)7-s + (0.142 − 0.896i)8-s + (−0.749 − 0.243i)9-s + (−0.810 − 0.706i)10-s + (−0.0505 + 0.0505i)12-s + (−0.389 − 0.764i)13-s + (0.0457 + 0.0629i)14-s + (−0.102 − 0.448i)15-s + (0.349 + 1.07i)16-s + (0.408 − 0.800i)17-s + (0.836 − 0.132i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.462052 - 0.143030i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.462052 - 0.143030i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.834 - 2.07i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (1.35 - 0.690i)T + (1.17 - 1.61i)T^{2} \) |
| 3 | \( 1 + (0.787 + 0.124i)T + (2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (0.0299 + 0.189i)T + (-6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (1.40 + 2.75i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.68 + 3.30i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-0.601 + 0.437i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (1.14 + 1.14i)T + 23iT^{2} \) |
| 29 | \( 1 + (7.72 + 5.61i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.108 + 0.333i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.33 + 0.845i)T + (35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (-3.93 - 5.41i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-3.72 + 3.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.93 + 12.2i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-8.04 + 4.09i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (5.65 - 7.78i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.60 + 1.82i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-4.13 + 4.13i)T - 67iT^{2} \) |
| 71 | \( 1 + (3.54 + 10.9i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.37 + 0.375i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-0.207 + 0.637i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (14.8 + 7.57i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 - 7.92iT - 89T^{2} \) |
| 97 | \( 1 + (0.626 + 1.22i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.33363117580165182073932829022, −9.702840820693370771550060354782, −8.863336036049716520890160593895, −7.76063481402133922125246571521, −7.20857006786480199792169550252, −6.19908841787593912433729505766, −5.44401938498131660048895643436, −3.77714733171708419014162868337, −2.58822384822137277987455489617, −0.43900884039905803293849535106,
1.21885539310661057109965789410, 2.41363442637777175140702671195, 4.26151103514692889369027859580, 5.39643817851721718811298461758, 5.91398230375174173760269332882, 7.51087272139693535808507066994, 8.429724101771743589946551511565, 9.134174070842294320798848516353, 9.731050902017970559849405412354, 10.71174055552759647301067210962