| L(s) = 1 | + 2-s + 6·5-s − 4·7-s − 8-s + 6·10-s + 6·11-s + 4·13-s − 4·14-s − 16-s + 4·19-s + 6·22-s + 17·25-s + 4·26-s − 9·29-s + 31-s − 24·35-s − 8·37-s + 4·38-s − 6·40-s + 10·43-s + 6·47-s + 9·49-s + 17·50-s + 3·53-s + 36·55-s + 4·56-s − 9·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.68·5-s − 1.51·7-s − 0.353·8-s + 1.89·10-s + 1.80·11-s + 1.10·13-s − 1.06·14-s − 1/4·16-s + 0.917·19-s + 1.27·22-s + 17/5·25-s + 0.784·26-s − 1.67·29-s + 0.179·31-s − 4.05·35-s − 1.31·37-s + 0.648·38-s − 0.948·40-s + 1.52·43-s + 0.875·47-s + 9/7·49-s + 2.40·50-s + 0.412·53-s + 4.85·55-s + 0.534·56-s − 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.336760248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.336760248\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−9.708486461572133428219338457256, −9.691122025135894266960511988428, −9.226637517113060471890790531188, −9.114712235258629449525666312509, −8.739549406789580451893183727713, −8.024947940780678758494062239078, −7.15498269762732117768440445654, −6.85418599755038843685713188580, −6.57932104836268738520748312337, −5.98305633488629024911144092968, −5.89224375491954948816210244283, −5.54156348627222818506458121028, −5.17560042390610385257426363034, −4.10870492650767046906267084903, −3.95234891995927076790138491199, −3.35661054123594239836902458680, −2.85087939988167470254451959724, −2.14775661198896399236058700745, −1.62767528143009047053710113133, −0.977203350482848606751121787607,
0.977203350482848606751121787607, 1.62767528143009047053710113133, 2.14775661198896399236058700745, 2.85087939988167470254451959724, 3.35661054123594239836902458680, 3.95234891995927076790138491199, 4.10870492650767046906267084903, 5.17560042390610385257426363034, 5.54156348627222818506458121028, 5.89224375491954948816210244283, 5.98305633488629024911144092968, 6.57932104836268738520748312337, 6.85418599755038843685713188580, 7.15498269762732117768440445654, 8.024947940780678758494062239078, 8.739549406789580451893183727713, 9.114712235258629449525666312509, 9.226637517113060471890790531188, 9.691122025135894266960511988428, 9.708486461572133428219338457256
