| L(s) = 1 | + 2·3-s + 2·9-s + 6·11-s − 6·13-s − 8·17-s − 2·19-s + 6·27-s − 6·29-s + 12·33-s − 6·37-s − 12·39-s − 6·43-s + 4·47-s + 14·49-s − 16·51-s − 18·53-s − 4·57-s + 18·59-s − 10·61-s + 6·67-s + 16·79-s + 11·81-s + 18·83-s − 12·87-s + 24·97-s + 12·99-s + 6·101-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2/3·9-s + 1.80·11-s − 1.66·13-s − 1.94·17-s − 0.458·19-s + 1.15·27-s − 1.11·29-s + 2.08·33-s − 0.986·37-s − 1.92·39-s − 0.914·43-s + 0.583·47-s + 2·49-s − 2.24·51-s − 2.47·53-s − 0.529·57-s + 2.34·59-s − 1.28·61-s + 0.733·67-s + 1.80·79-s + 11/9·81-s + 1.97·83-s − 1.28·87-s + 2.43·97-s + 1.20·99-s + 0.597·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.796823263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.796823263\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
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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
−9.297979972482170517350847655877, −9.130595015766078544735994199816, −8.924500959913202220969583298190, −8.669748675047618226351140687924, −7.87548490829461567352717752556, −7.82161665941878416819812405175, −7.15455050290143384610473330998, −6.69932726753547858786528710932, −6.69160712520191154728844575589, −6.16393250749110025375667750850, −5.39656596233904360094336024295, −4.92381506826448077951628032774, −4.50809838337175622411910204168, −4.13367813785045138142373593225, −3.58516211411290056779105518531, −3.25365281606540431105998969046, −2.35708580210966220561969062932, −2.21221419329512794238587704599, −1.67782821992668627847569775687, −0.58016147763625966245952406500,
0.58016147763625966245952406500, 1.67782821992668627847569775687, 2.21221419329512794238587704599, 2.35708580210966220561969062932, 3.25365281606540431105998969046, 3.58516211411290056779105518531, 4.13367813785045138142373593225, 4.50809838337175622411910204168, 4.92381506826448077951628032774, 5.39656596233904360094336024295, 6.16393250749110025375667750850, 6.69160712520191154728844575589, 6.69932726753547858786528710932, 7.15455050290143384610473330998, 7.82161665941878416819812405175, 7.87548490829461567352717752556, 8.669748675047618226351140687924, 8.924500959913202220969583298190, 9.130595015766078544735994199816, 9.297979972482170517350847655877
