| L(s) = 1 | − 2·3-s + 4·5-s − 2·7-s + 3·9-s − 4·13-s − 8·15-s − 2·17-s + 4·21-s + 8·23-s + 2·25-s − 4·27-s + 4·29-s − 8·35-s − 4·37-s + 8·39-s − 4·41-s − 8·43-s + 12·45-s + 16·47-s + 3·49-s + 4·51-s + 12·53-s + 8·59-s + 20·61-s − 6·63-s − 16·65-s + 8·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s − 0.755·7-s + 9-s − 1.10·13-s − 2.06·15-s − 0.485·17-s + 0.872·21-s + 1.66·23-s + 2/5·25-s − 0.769·27-s + 0.742·29-s − 1.35·35-s − 0.657·37-s + 1.28·39-s − 0.624·41-s − 1.21·43-s + 1.78·45-s + 2.33·47-s + 3/7·49-s + 0.560·51-s + 1.64·53-s + 1.04·59-s + 2.56·61-s − 0.755·63-s − 1.98·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32626944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32626944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.412378217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.412378217\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 24 T + 286 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 86 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−8.401623292797585664846043336208, −7.88410839683485929838060750231, −7.30355865638477833800684086370, −7.03678399563654217145024819803, −6.83634885094482850788032048503, −6.51738791382209598532333492140, −6.20226460058585050195656426300, −5.65248135210590006836623387310, −5.37695213827609845087914980861, −5.36781726853962615356005390727, −4.84468347416322590841025997214, −4.45085121858457059840735139381, −3.81614883294576252887817220802, −3.61180129386281851974508791446, −2.74040109881289772290551498915, −2.57326782856774654495500389038, −2.06218888505628783581297165438, −1.69942441547186803777852439627, −0.864076720247538904497898252984, −0.54188328923494244407794846359,
0.54188328923494244407794846359, 0.864076720247538904497898252984, 1.69942441547186803777852439627, 2.06218888505628783581297165438, 2.57326782856774654495500389038, 2.74040109881289772290551498915, 3.61180129386281851974508791446, 3.81614883294576252887817220802, 4.45085121858457059840735139381, 4.84468347416322590841025997214, 5.36781726853962615356005390727, 5.37695213827609845087914980861, 5.65248135210590006836623387310, 6.20226460058585050195656426300, 6.51738791382209598532333492140, 6.83634885094482850788032048503, 7.03678399563654217145024819803, 7.30355865638477833800684086370, 7.88410839683485929838060750231, 8.401623292797585664846043336208
