| L(s) = 1 | + 8·11-s + 2·19-s + 8·29-s + 10·31-s − 24·41-s + 13·49-s − 16·59-s + 14·61-s + 24·71-s + 24·79-s − 24·101-s + 14·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2.41·11-s + 0.458·19-s + 1.48·29-s + 1.79·31-s − 3.74·41-s + 13/7·49-s − 2.08·59-s + 1.79·61-s + 2.84·71-s + 2.70·79-s − 2.38·101-s + 1.34·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.177401292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.177401292\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−8.559329833718780737585645084983, −8.475228181135524051090930607721, −8.091134967259805296762080742884, −7.76455836642957990418026054521, −6.90930272337620533188762753234, −6.85126321159588902687012436004, −6.51902654231889450183513660349, −6.47840349335689484587310372512, −5.67605902909172650935027472209, −5.42225340581984716543995904422, −4.76283793191440230393048767946, −4.65734777800103942978237311443, −3.96169618313518412497596125228, −3.83167034023701742621503005026, −3.11997496295862470772801122941, −3.04505998799951721824141860619, −2.04605069918607462848912625190, −1.82632277824063517047850333270, −0.978429553220906821166989932321, −0.78241731799460590624972547120,
0.78241731799460590624972547120, 0.978429553220906821166989932321, 1.82632277824063517047850333270, 2.04605069918607462848912625190, 3.04505998799951721824141860619, 3.11997496295862470772801122941, 3.83167034023701742621503005026, 3.96169618313518412497596125228, 4.65734777800103942978237311443, 4.76283793191440230393048767946, 5.42225340581984716543995904422, 5.67605902909172650935027472209, 6.47840349335689484587310372512, 6.51902654231889450183513660349, 6.85126321159588902687012436004, 6.90930272337620533188762753234, 7.76455836642957990418026054521, 8.091134967259805296762080742884, 8.475228181135524051090930607721, 8.559329833718780737585645084983
