L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.258 − 0.965i)5-s + (1.36 + 0.366i)7-s + (0.707 − 0.707i)8-s + 10-s + (1.22 − 0.707i)11-s + (−0.707 + 1.22i)14-s + (0.500 + 0.866i)16-s + (−0.258 + 0.965i)20-s + (0.366 + 1.36i)22-s + (−0.866 + 0.499i)25-s + (−1 − 0.999i)28-s + (−0.707 − 1.22i)29-s + (−0.965 + 0.258i)32-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.258 − 0.965i)5-s + (1.36 + 0.366i)7-s + (0.707 − 0.707i)8-s + 10-s + (1.22 − 0.707i)11-s + (−0.707 + 1.22i)14-s + (0.500 + 0.866i)16-s + (−0.258 + 0.965i)20-s + (0.366 + 1.36i)22-s + (−0.866 + 0.499i)25-s + (−1 − 0.999i)28-s + (−0.707 − 1.22i)29-s + (−0.965 + 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.194820459\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.194820459\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.258 + 0.965i)T \) |
good | 7 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
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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
−8.668094474357129826565991536631, −8.169136085108722215946123785104, −7.59174362540973355183532279851, −6.58582236391490149320659098234, −5.76090556771633340221420016810, −5.17660448040073734751947823337, −4.37747213883342184620376687466, −3.76384839486532364638396405280, −1.88526767513107040337032500074, −0.951242879217006353478242309300,
1.36247416261263244052051069273, 2.08985847665928808487871121186, 3.23515186535450493672878428001, 4.06332290952289298815351399802, 4.61633775901056302953269354903, 5.63451102091095697148742304451, 6.91163875239270216766228730330, 7.36915697470399411714988893022, 8.171776874624366517055217700336, 8.878502579450949482406167828274