| L(s) = 1 | − 4-s + 6·5-s − 4·7-s − 3·9-s − 6·13-s + 16-s − 6·20-s + 17·25-s + 4·28-s − 24·35-s + 3·36-s − 18·45-s − 2·49-s + 6·52-s + 2·53-s − 8·59-s + 12·63-s − 64-s − 36·65-s + 8·67-s + 4·71-s + 6·80-s + 24·91-s − 17·100-s − 12·103-s − 4·107-s − 2·109-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2.68·5-s − 1.51·7-s − 9-s − 1.66·13-s + 1/4·16-s − 1.34·20-s + 17/5·25-s + 0.755·28-s − 4.05·35-s + 1/2·36-s − 2.68·45-s − 2/7·49-s + 0.832·52-s + 0.274·53-s − 1.04·59-s + 1.51·63-s − 1/8·64-s − 4.46·65-s + 0.977·67-s + 0.474·71-s + 0.670·80-s + 2.51·91-s − 1.69·100-s − 1.18·103-s − 0.386·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2829124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2829124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.363983389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.363983389\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 29 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 21 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 158 T^{2} + p^{2} T^{4} \) |
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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
−9.722069607411543082928815530656, −9.443036635129054403487352549361, −8.984475582426802165078203081386, −8.566060319151987108241448223752, −8.018388553981976775876385309011, −7.56525463191707240072684328125, −6.87587112949016873299982435127, −6.68014073435152539812124297049, −6.25288933971183422438460006396, −5.84607367904095129678844366097, −5.62369630595312297732320791867, −5.13188491016681541977279145476, −4.87616465118659609228856566843, −4.15947300507508592625729898362, −3.41691704988947794663062719915, −2.91250706337636848359381278780, −2.60971144266374629866654058573, −2.11057303024008854148953518259, −1.54415424273204574331276588812, −0.41981258853431780794457895063,
0.41981258853431780794457895063, 1.54415424273204574331276588812, 2.11057303024008854148953518259, 2.60971144266374629866654058573, 2.91250706337636848359381278780, 3.41691704988947794663062719915, 4.15947300507508592625729898362, 4.87616465118659609228856566843, 5.13188491016681541977279145476, 5.62369630595312297732320791867, 5.84607367904095129678844366097, 6.25288933971183422438460006396, 6.68014073435152539812124297049, 6.87587112949016873299982435127, 7.56525463191707240072684328125, 8.018388553981976775876385309011, 8.566060319151987108241448223752, 8.984475582426802165078203081386, 9.443036635129054403487352549361, 9.722069607411543082928815530656
