| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 4·13-s + 5·16-s + 4·19-s + 6·23-s − 10·25-s + 8·26-s + 10·31-s + 6·32-s − 8·37-s + 8·38-s + 6·41-s + 4·43-s + 12·46-s + 18·47-s − 20·50-s + 12·52-s + 4·61-s + 20·62-s + 7·64-s − 8·67-s − 6·71-s − 14·73-s − 16·74-s + 12·76-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 1.10·13-s + 5/4·16-s + 0.917·19-s + 1.25·23-s − 2·25-s + 1.56·26-s + 1.79·31-s + 1.06·32-s − 1.31·37-s + 1.29·38-s + 0.937·41-s + 0.609·43-s + 1.76·46-s + 2.62·47-s − 2.82·50-s + 1.66·52-s + 0.512·61-s + 2.54·62-s + 7/8·64-s − 0.977·67-s − 0.712·71-s − 1.63·73-s − 1.85·74-s + 1.37·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.78863573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.78863573\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 157 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 88 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 108 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 133 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 165 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 292 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 115 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 138 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.70161476514405457012856076169, −7.64881818274005793129962247361, −7.24369061841218580801201993808, −7.00771464529860782769396660494, −6.41217437750548102955904505747, −6.14816792832787184383632766563, −5.89066765527308498626847438714, −5.69018829035559722487623337642, −5.16219504112330216213752204497, −4.83137922198848655727291256855, −4.52313541151255672851331147525, −4.05906611938360777203433040467, −3.66723363976203350272019754494, −3.53330106696162525732118237625, −2.96386756593487151351751450024, −2.60144274372572179159405572247, −2.18301619449332407510495761933, −1.66509796729960218976711994202, −1.05337508707568473600596514998, −0.69887583669390081128176484699,
0.69887583669390081128176484699, 1.05337508707568473600596514998, 1.66509796729960218976711994202, 2.18301619449332407510495761933, 2.60144274372572179159405572247, 2.96386756593487151351751450024, 3.53330106696162525732118237625, 3.66723363976203350272019754494, 4.05906611938360777203433040467, 4.52313541151255672851331147525, 4.83137922198848655727291256855, 5.16219504112330216213752204497, 5.69018829035559722487623337642, 5.89066765527308498626847438714, 6.14816792832787184383632766563, 6.41217437750548102955904505747, 7.00771464529860782769396660494, 7.24369061841218580801201993808, 7.64881818274005793129962247361, 7.70161476514405457012856076169
