L(s) = 1 | + (−0.798 − 0.602i)2-s + (0.273 + 0.961i)4-s + (−0.739 − 0.326i)5-s + (0.361 − 0.932i)8-s + (−0.995 + 0.0922i)9-s + (0.393 + 0.705i)10-s + (1.85 + 0.719i)13-s + (−0.850 + 0.526i)16-s + (0.850 − 0.526i)17-s + (0.850 + 0.526i)18-s + (0.111 − 0.800i)20-s + (−0.234 − 0.256i)25-s + (−1.04 − 1.69i)26-s + (0.748 − 1.34i)29-s + (0.995 + 0.0922i)32-s + ⋯ |
L(s) = 1 | + (−0.798 − 0.602i)2-s + (0.273 + 0.961i)4-s + (−0.739 − 0.326i)5-s + (0.361 − 0.932i)8-s + (−0.995 + 0.0922i)9-s + (0.393 + 0.705i)10-s + (1.85 + 0.719i)13-s + (−0.850 + 0.526i)16-s + (0.850 − 0.526i)17-s + (0.850 + 0.526i)18-s + (0.111 − 0.800i)20-s + (−0.234 − 0.256i)25-s + (−1.04 − 1.69i)26-s + (0.748 − 1.34i)29-s + (0.995 + 0.0922i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 + 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6339641708\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6339641708\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.798 + 0.602i)T \) |
| 17 | \( 1 + (-0.850 + 0.526i)T \) |
good | 3 | \( 1 + (0.995 - 0.0922i)T^{2} \) |
| 5 | \( 1 + (0.739 + 0.326i)T + (0.673 + 0.739i)T^{2} \) |
| 7 | \( 1 + (-0.183 - 0.982i)T^{2} \) |
| 11 | \( 1 + (-0.526 - 0.850i)T^{2} \) |
| 13 | \( 1 + (-1.85 - 0.719i)T + (0.739 + 0.673i)T^{2} \) |
| 19 | \( 1 + (-0.273 + 0.961i)T^{2} \) |
| 23 | \( 1 + (-0.183 - 0.982i)T^{2} \) |
| 29 | \( 1 + (-0.748 + 1.34i)T + (-0.526 - 0.850i)T^{2} \) |
| 31 | \( 1 + (0.673 - 0.739i)T^{2} \) |
| 37 | \( 1 + (-0.0878 + 0.261i)T + (-0.798 - 0.602i)T^{2} \) |
| 41 | \( 1 + (-1.82 + 0.0844i)T + (0.995 - 0.0922i)T^{2} \) |
| 43 | \( 1 + (0.445 + 0.895i)T^{2} \) |
| 47 | \( 1 + (0.982 + 0.183i)T^{2} \) |
| 53 | \( 1 + (0.365 - 0.0339i)T + (0.982 - 0.183i)T^{2} \) |
| 59 | \( 1 + (0.932 + 0.361i)T^{2} \) |
| 61 | \( 1 + (-1.12 + 1.64i)T + (-0.361 - 0.932i)T^{2} \) |
| 67 | \( 1 + (0.273 - 0.961i)T^{2} \) |
| 71 | \( 1 + (0.183 + 0.982i)T^{2} \) |
| 73 | \( 1 + (-1.24 - 0.292i)T + (0.895 + 0.445i)T^{2} \) |
| 79 | \( 1 + (-0.961 - 0.273i)T^{2} \) |
| 83 | \( 1 + (0.0922 - 0.995i)T^{2} \) |
| 89 | \( 1 + (1.48 - 0.576i)T + (0.739 - 0.673i)T^{2} \) |
| 97 | \( 1 + (1.11 - 1.34i)T + (-0.183 - 0.982i)T^{2} \) |
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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
−9.726114792669753749951123057497, −9.066830204667653690111672252195, −8.170984036284369830007865730240, −7.952178372837805739437744112668, −6.65055715642403007885931433444, −5.78949430907393889040949356566, −4.30987277144579036828123708367, −3.59778504177246814750413860371, −2.50216767731928666820568373521, −0.957098284840174723029601516978,
1.16001575000915243741753499127, 2.93581227350271891311999678964, 3.87956989293361627405368327746, 5.41828177346288504166890234948, 5.96961350214367625648507248427, 6.87600688482695432594246717885, 7.888994622524675584832884243841, 8.367277093555646216971999039851, 9.002163258129608897045261947377, 10.11725682798887949674698349012