| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 4·9-s + 5·16-s − 4·17-s − 8·18-s + 8·19-s + 8·23-s − 2·25-s + 8·29-s + 8·31-s + 6·32-s − 8·34-s − 12·36-s + 16·38-s + 2·41-s + 8·43-s + 16·46-s + 4·47-s − 4·50-s + 24·53-s + 16·58-s + 8·59-s − 12·61-s + 16·62-s + 7·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 4/3·9-s + 5/4·16-s − 0.970·17-s − 1.88·18-s + 1.83·19-s + 1.66·23-s − 2/5·25-s + 1.48·29-s + 1.43·31-s + 1.06·32-s − 1.37·34-s − 2·36-s + 2.59·38-s + 0.312·41-s + 1.21·43-s + 2.35·46-s + 0.583·47-s − 0.565·50-s + 3.29·53-s + 2.10·58-s + 1.04·59-s − 1.53·61-s + 2.03·62-s + 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16144324 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.798990405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.798990405\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 42 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 144 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 176 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.664339287645167579210275064160, −8.419208831928178674758679746672, −7.54051509895902645996352062928, −7.35865150360153649197503938395, −7.32070355252894300751584786239, −6.67668138471615580985756980990, −6.17220209743256085656077188515, −6.03996967397681996855914135167, −5.66231011692768892111672555028, −5.22353021049217079469617768109, −4.78171927277246922727363290691, −4.68240750706451069563245159426, −4.07597582265702009364796425906, −3.63596680261222658258079574499, −3.06796514640028810599241340590, −2.88590782453908926574169422502, −2.50219407708370172621868696794, −2.05465533367107995324406273490, −0.972113670146921169218170205783, −0.832630225570266192705647946280,
0.832630225570266192705647946280, 0.972113670146921169218170205783, 2.05465533367107995324406273490, 2.50219407708370172621868696794, 2.88590782453908926574169422502, 3.06796514640028810599241340590, 3.63596680261222658258079574499, 4.07597582265702009364796425906, 4.68240750706451069563245159426, 4.78171927277246922727363290691, 5.22353021049217079469617768109, 5.66231011692768892111672555028, 6.03996967397681996855914135167, 6.17220209743256085656077188515, 6.67668138471615580985756980990, 7.32070355252894300751584786239, 7.35865150360153649197503938395, 7.54051509895902645996352062928, 8.419208831928178674758679746672, 8.664339287645167579210275064160
