| L(s) = 1 | + 2·5-s + 4·7-s − 11-s − 13-s + 3·17-s + 19-s − 8·23-s − 25-s + 6·29-s + 8·35-s − 8·37-s − 8·43-s − 2·47-s + 9·49-s + 9·53-s − 2·55-s + 3·59-s − 10·61-s − 2·65-s + 4·67-s + 3·71-s + 4·73-s − 4·77-s − 15·79-s − 9·83-s + 6·85-s + 89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 0.301·11-s − 0.277·13-s + 0.727·17-s + 0.229·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s + 1.35·35-s − 1.31·37-s − 1.21·43-s − 0.291·47-s + 9/7·49-s + 1.23·53-s − 0.269·55-s + 0.390·59-s − 1.28·61-s − 0.248·65-s + 0.488·67-s + 0.356·71-s + 0.468·73-s − 0.455·77-s − 1.68·79-s − 0.987·83-s + 0.650·85-s + 0.105·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.92144072412485, −13.52436153728840, −12.84532322033845, −12.24938084554298, −11.75752037027418, −11.62785241261832, −10.78034619722973, −10.28140082371957, −10.06809854919871, −9.538307964637885, −8.794653501626417, −8.299093659827024, −8.028344709318493, −7.429321174994899, −6.868035762147757, −6.210634093833655, −5.585553002102414, −5.346461183648019, −4.733344840692315, −4.215804029595441, −3.503791865998088, −2.763139400444831, −2.057611618696621, −1.715565707426906, −1.076116898443193, 0,
1.076116898443193, 1.715565707426906, 2.057611618696621, 2.763139400444831, 3.503791865998088, 4.215804029595441, 4.733344840692315, 5.346461183648019, 5.585553002102414, 6.210634093833655, 6.868035762147757, 7.429321174994899, 8.028344709318493, 8.299093659827024, 8.794653501626417, 9.538307964637885, 10.06809854919871, 10.28140082371957, 10.78034619722973, 11.62785241261832, 11.75752037027418, 12.24938084554298, 12.84532322033845, 13.52436153728840, 13.92144072412485
