L(s) = 1 | − 3-s + 6·5-s + 7-s − 6·11-s − 6·15-s + 5·19-s − 21-s + 12·23-s + 19·25-s + 27-s − 5·31-s + 6·33-s + 6·35-s + 11·37-s − 6·47-s − 6·49-s − 12·53-s − 36·55-s − 5·57-s + 12·59-s − 24·61-s + 15·67-s − 12·69-s − 9·73-s − 19·75-s − 6·77-s + 21·79-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 2.68·5-s + 0.377·7-s − 1.80·11-s − 1.54·15-s + 1.14·19-s − 0.218·21-s + 2.50·23-s + 19/5·25-s + 0.192·27-s − 0.898·31-s + 1.04·33-s + 1.01·35-s + 1.80·37-s − 0.875·47-s − 6/7·49-s − 1.64·53-s − 4.85·55-s − 0.662·57-s + 1.56·59-s − 3.07·61-s + 1.83·67-s − 1.44·69-s − 1.05·73-s − 2.19·75-s − 0.683·77-s + 2.36·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.098084052\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.098084052\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.48059601693695082287258680054, −11.20943870587854103703984502365, −10.86450022854068465925634146344, −10.36881206496366187664975706833, −9.853340240000326350911008598417, −9.682911273098404974947397808048, −9.096739429209811744925015113418, −8.786492370745372079970364365543, −7.78970096506374635205014125970, −7.63642700615454250600018195831, −6.75650839165848230290535772432, −6.38237499223813147705628138950, −5.75670843772835619177813447335, −5.40016266680739167177379883499, −5.06096780785574179146491865054, −4.69264177783429465070703001545, −3.08164935446251571575450347939, −2.81042278818642328523984568290, −1.97608912638688449030676923539, −1.19406582918638195211694067675,
1.19406582918638195211694067675, 1.97608912638688449030676923539, 2.81042278818642328523984568290, 3.08164935446251571575450347939, 4.69264177783429465070703001545, 5.06096780785574179146491865054, 5.40016266680739167177379883499, 5.75670843772835619177813447335, 6.38237499223813147705628138950, 6.75650839165848230290535772432, 7.63642700615454250600018195831, 7.78970096506374635205014125970, 8.786492370745372079970364365543, 9.096739429209811744925015113418, 9.682911273098404974947397808048, 9.853340240000326350911008598417, 10.36881206496366187664975706833, 10.86450022854068465925634146344, 11.20943870587854103703984502365, 11.48059601693695082287258680054