L(s) = 1 | + 2-s − 4-s − 5-s + 7-s − 3·8-s − 10-s + 2·11-s + 2·13-s + 14-s − 16-s − 2·19-s + 20-s + 2·22-s − 2·23-s + 25-s + 2·26-s − 28-s + 6·29-s − 8·31-s + 5·32-s − 35-s − 8·37-s − 2·38-s + 3·40-s + 2·41-s + 4·43-s − 2·44-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s − 1.06·8-s − 0.316·10-s + 0.603·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.458·19-s + 0.223·20-s + 0.426·22-s − 0.417·23-s + 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.11·29-s − 1.43·31-s + 0.883·32-s − 0.169·35-s − 1.31·37-s − 0.324·38-s + 0.474·40-s + 0.312·41-s + 0.609·43-s − 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.07304544599635, −13.78920537222167, −12.94359703441174, −12.80619316698651, −12.13413393479484, −11.84592400475571, −11.17272723838625, −10.83481335791518, −10.14318914331507, −9.568956822914341, −9.020951815424139, −8.589489458642464, −8.170353043657987, −7.596869091257785, −6.753167161042896, −6.503057394906470, −5.769596321006436, −5.234138376143351, −4.793776384414195, −4.012362122792747, −3.853263007597552, −3.225649884000020, −2.441957133560242, −1.656183004858821, −0.8559079650246240, 0,
0.8559079650246240, 1.656183004858821, 2.441957133560242, 3.225649884000020, 3.853263007597552, 4.012362122792747, 4.793776384414195, 5.234138376143351, 5.769596321006436, 6.503057394906470, 6.753167161042896, 7.596869091257785, 8.170353043657987, 8.589489458642464, 9.020951815424139, 9.568956822914341, 10.14318914331507, 10.83481335791518, 11.17272723838625, 11.84592400475571, 12.13413393479484, 12.80619316698651, 12.94359703441174, 13.78920537222167, 14.07304544599635