| L(s) = 1 | − 2·5-s + 11-s + 6·13-s − 17-s − 5·19-s + 7·23-s − 25-s + 29-s − 2·31-s + 3·37-s + 2·41-s − 43-s + 7·47-s − 8·53-s − 2·55-s − 3·59-s − 2·61-s − 12·65-s − 3·71-s + 4·73-s − 8·79-s − 6·83-s + 2·85-s − 10·89-s + 10·95-s + 7·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.301·11-s + 1.66·13-s − 0.242·17-s − 1.14·19-s + 1.45·23-s − 1/5·25-s + 0.185·29-s − 0.359·31-s + 0.493·37-s + 0.312·41-s − 0.152·43-s + 1.02·47-s − 1.09·53-s − 0.269·55-s − 0.390·59-s − 0.256·61-s − 1.48·65-s − 0.356·71-s + 0.468·73-s − 0.900·79-s − 0.658·83-s + 0.216·85-s − 1.05·89-s + 1.02·95-s + 0.710·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.23638054020027, −14.63542385932846, −14.05532305006808, −13.43907191821173, −13.00079160611976, −12.52979452779777, −11.90356982644083, −11.28335124049836, −10.93508541005598, −10.65258197163061, −9.715752844661756, −9.105158998527003, −8.621356966947669, −8.245253422419325, −7.573935918023176, −6.972605105066197, −6.390738970354100, −5.914434861721923, −5.151736464205967, −4.265372741031757, −4.076855338022968, −3.321668100547862, −2.695924023760142, −1.667033591816374, −0.9944420260601132, 0,
0.9944420260601132, 1.667033591816374, 2.695924023760142, 3.321668100547862, 4.076855338022968, 4.265372741031757, 5.151736464205967, 5.914434861721923, 6.390738970354100, 6.972605105066197, 7.573935918023176, 8.245253422419325, 8.621356966947669, 9.105158998527003, 9.715752844661756, 10.65258197163061, 10.93508541005598, 11.28335124049836, 11.90356982644083, 12.52979452779777, 13.00079160611976, 13.43907191821173, 14.05532305006808, 14.63542385932846, 15.23638054020027
