| L(s) = 1 | − 4·5-s + 5·7-s − 6·13-s + 4·17-s − 7·19-s − 4·23-s + 5·25-s − 16·29-s + 5·31-s − 20·35-s − 3·37-s + 16·41-s + 22·43-s − 4·47-s + 18·49-s − 4·53-s − 12·59-s + 2·61-s + 24·65-s + 3·67-s + 24·71-s − 73-s − 79-s + 24·83-s − 16·85-s − 8·89-s − 30·91-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.88·7-s − 1.66·13-s + 0.970·17-s − 1.60·19-s − 0.834·23-s + 25-s − 2.97·29-s + 0.898·31-s − 3.38·35-s − 0.493·37-s + 2.49·41-s + 3.35·43-s − 0.583·47-s + 18/7·49-s − 0.549·53-s − 1.56·59-s + 0.256·61-s + 2.97·65-s + 0.366·67-s + 2.84·71-s − 0.117·73-s − 0.112·79-s + 2.63·83-s − 1.73·85-s − 0.847·89-s − 3.14·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.076837398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.076837398\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.98924161734576861982342676356, −10.90159824313661073380379620450, −10.62010952116005021816092700862, −9.637049345552500459431017087019, −9.385722123568035517305389712031, −8.915407422879308146679300507267, −7.966977003140255282520592127949, −7.927506598256312011528044874377, −7.82275404533879712100488741639, −7.36748687500855343626696439633, −6.77641620994493579268133348599, −5.70776048176978650128831836495, −5.66617926374476197732742632562, −4.77501246160384940741830682600, −4.24230761939033025582636775265, −4.21128446977269884399013820835, −3.46278771705492276316665914800, −2.32856040431564032607720466055, −2.02300429895265815354099123939, −0.61964257847577541227096568790,
0.61964257847577541227096568790, 2.02300429895265815354099123939, 2.32856040431564032607720466055, 3.46278771705492276316665914800, 4.21128446977269884399013820835, 4.24230761939033025582636775265, 4.77501246160384940741830682600, 5.66617926374476197732742632562, 5.70776048176978650128831836495, 6.77641620994493579268133348599, 7.36748687500855343626696439633, 7.82275404533879712100488741639, 7.927506598256312011528044874377, 7.966977003140255282520592127949, 8.915407422879308146679300507267, 9.385722123568035517305389712031, 9.637049345552500459431017087019, 10.62010952116005021816092700862, 10.90159824313661073380379620450, 10.98924161734576861982342676356
