| L(s) = 1 | − 2·5-s + 4·7-s + 4·11-s − 4·13-s + 8·17-s − 4·19-s + 4·23-s + 3·25-s + 10·29-s − 4·31-s − 8·35-s + 6·41-s − 16·43-s + 4·47-s + 49-s + 12·53-s − 8·55-s + 8·59-s − 10·61-s + 8·65-s + 16·67-s + 12·71-s + 16·77-s + 24·79-s − 16·83-s − 16·85-s − 6·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s + 1.20·11-s − 1.10·13-s + 1.94·17-s − 0.917·19-s + 0.834·23-s + 3/5·25-s + 1.85·29-s − 0.718·31-s − 1.35·35-s + 0.937·41-s − 2.43·43-s + 0.583·47-s + 1/7·49-s + 1.64·53-s − 1.07·55-s + 1.04·59-s − 1.28·61-s + 0.992·65-s + 1.95·67-s + 1.42·71-s + 1.82·77-s + 2.70·79-s − 1.75·83-s − 1.73·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.385377634\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.385377634\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 47 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 95 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 26 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 135 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 195 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 290 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 203 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 139 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.539032675421461652556754304434, −8.411695803328142220955323896520, −8.162696644466281442980506775081, −7.85537240026844368451019320721, −7.26930211378005127087667382528, −7.14619538616109604002998026027, −6.53051922064057432402288161963, −6.52226565421884555733939599773, −5.49649462228231899873400592898, −5.48008761568390059159659218174, −4.80908709242101333128389554160, −4.80178090510442437798438678220, −4.06475336630396918571344646447, −3.94000709814917147714044377279, −3.26658528621540987784167707251, −2.93492855673892803720948024033, −2.21457265681049892157422747490, −1.74528450056598654506904594834, −1.08643187881955290246124910591, −0.67541375512535730797162074079,
0.67541375512535730797162074079, 1.08643187881955290246124910591, 1.74528450056598654506904594834, 2.21457265681049892157422747490, 2.93492855673892803720948024033, 3.26658528621540987784167707251, 3.94000709814917147714044377279, 4.06475336630396918571344646447, 4.80178090510442437798438678220, 4.80908709242101333128389554160, 5.48008761568390059159659218174, 5.49649462228231899873400592898, 6.52226565421884555733939599773, 6.53051922064057432402288161963, 7.14619538616109604002998026027, 7.26930211378005127087667382528, 7.85537240026844368451019320721, 8.162696644466281442980506775081, 8.411695803328142220955323896520, 8.539032675421461652556754304434
