| L(s) = 1 | − 5-s + 4·11-s + 6·13-s + 2·17-s + 25-s − 6·29-s + 8·31-s − 10·37-s + 2·41-s − 4·43-s − 8·47-s + 2·53-s − 4·55-s + 8·59-s + 14·61-s − 6·65-s + 12·67-s − 16·71-s − 2·73-s + 8·79-s − 8·83-s − 2·85-s + 10·89-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s + 1.66·13-s + 0.485·17-s + 1/5·25-s − 1.11·29-s + 1.43·31-s − 1.64·37-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 0.274·53-s − 0.539·55-s + 1.04·59-s + 1.79·61-s − 0.744·65-s + 1.46·67-s − 1.89·71-s − 0.234·73-s + 0.900·79-s − 0.878·83-s − 0.216·85-s + 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.736226644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.736226644\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.82158052079990, −14.51330819182431, −13.86979764511334, −13.39197365347187, −12.90766581414440, −12.21137684816247, −11.64677768007785, −11.40789149982676, −10.80914501462042, −10.07929547903235, −9.689790491285668, −8.784240186475920, −8.607175719134478, −8.055029744597415, −7.221044750048069, −6.744639592297828, −6.197138701373465, −5.623121964390526, −4.884463583365661, −4.115164767942684, −3.593166264557916, −3.256210638966484, −2.068268469107738, −1.361137803827452, −0.6768813151313336,
0.6768813151313336, 1.361137803827452, 2.068268469107738, 3.256210638966484, 3.593166264557916, 4.115164767942684, 4.884463583365661, 5.623121964390526, 6.197138701373465, 6.744639592297828, 7.221044750048069, 8.055029744597415, 8.607175719134478, 8.784240186475920, 9.689790491285668, 10.07929547903235, 10.80914501462042, 11.40789149982676, 11.64677768007785, 12.21137684816247, 12.90766581414440, 13.39197365347187, 13.86979764511334, 14.51330819182431, 14.82158052079990
