L(s) = 1 | + (0.739 − 0.673i)2-s + (0.0922 − 0.995i)4-s + (0.726 − 0.961i)5-s + (−0.602 − 0.798i)8-s + (−0.850 − 0.526i)9-s + (−0.111 − 1.20i)10-s + (1.02 + 1.35i)13-s + (−0.982 − 0.183i)16-s + (−0.982 − 0.183i)17-s + (−0.982 + 0.183i)18-s + (−0.890 − 0.811i)20-s + (−0.123 − 0.435i)25-s + (1.67 + 0.312i)26-s + (−0.156 + 1.69i)29-s + (−0.850 + 0.526i)32-s + ⋯ |
L(s) = 1 | + (0.739 − 0.673i)2-s + (0.0922 − 0.995i)4-s + (0.726 − 0.961i)5-s + (−0.602 − 0.798i)8-s + (−0.850 − 0.526i)9-s + (−0.111 − 1.20i)10-s + (1.02 + 1.35i)13-s + (−0.982 − 0.183i)16-s + (−0.982 − 0.183i)17-s + (−0.982 + 0.183i)18-s + (−0.890 − 0.811i)20-s + (−0.123 − 0.435i)25-s + (1.67 + 0.312i)26-s + (−0.156 + 1.69i)29-s + (−0.850 + 0.526i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.609232098\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.609232098\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.739 + 0.673i)T \) |
| 17 | \( 1 + (0.982 + 0.183i)T \) |
good | 3 | \( 1 + (0.850 + 0.526i)T^{2} \) |
| 5 | \( 1 + (-0.726 + 0.961i)T + (-0.273 - 0.961i)T^{2} \) |
| 7 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 11 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 13 | \( 1 + (-1.02 - 1.35i)T + (-0.273 + 0.961i)T^{2} \) |
| 19 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 23 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 29 | \( 1 + (0.156 - 1.69i)T + (-0.982 - 0.183i)T^{2} \) |
| 31 | \( 1 + (0.273 - 0.961i)T^{2} \) |
| 37 | \( 1 + (-1.37 + 0.533i)T + (0.739 - 0.673i)T^{2} \) |
| 41 | \( 1 + (-0.329 + 1.15i)T + (-0.850 - 0.526i)T^{2} \) |
| 43 | \( 1 + (-0.932 + 0.361i)T^{2} \) |
| 47 | \( 1 + (-0.445 + 0.895i)T^{2} \) |
| 53 | \( 1 + (0.757 + 0.469i)T + (0.445 + 0.895i)T^{2} \) |
| 59 | \( 1 + (0.602 + 0.798i)T^{2} \) |
| 61 | \( 1 + (0.243 + 0.489i)T + (-0.602 + 0.798i)T^{2} \) |
| 67 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 71 | \( 1 + (-0.445 - 0.895i)T^{2} \) |
| 73 | \( 1 + (-1.67 - 0.312i)T + (0.932 + 0.361i)T^{2} \) |
| 79 | \( 1 + (-0.0922 - 0.995i)T^{2} \) |
| 83 | \( 1 + (0.850 - 0.526i)T^{2} \) |
| 89 | \( 1 + (0.890 - 1.17i)T + (-0.273 - 0.961i)T^{2} \) |
| 97 | \( 1 + (0.156 + 0.0971i)T + (0.445 + 0.895i)T^{2} \) |
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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
−9.533653798966588652708664903371, −9.153321119254892012765349119496, −8.572413710615375542941620030204, −6.88461374118163339951368401760, −6.15268344634926004126956112270, −5.41819275404912229925939063355, −4.51687446033115446550327774914, −3.63178619744009745386585385418, −2.34222585378623043777751110872, −1.28601908529817087469629756301,
2.40560666274826302324379782136, 3.03329303020964136862749841118, 4.21048304089933465495987291443, 5.39262543540979118941910256862, 6.10918117660060455511329197471, 6.52890189443163708210717118666, 7.84056742063706530218493863753, 8.223974109974248530842827269904, 9.312327733683215623345904970372, 10.40400539055841316805212408243