| L(s) = 1 | − 4·5-s − 4·7-s + 4·11-s + 4·17-s + 8·23-s + 4·25-s − 4·29-s − 12·31-s + 16·35-s − 8·37-s + 12·41-s + 8·43-s + 8·47-s − 4·53-s − 16·55-s − 8·59-s − 8·61-s + 16·67-s + 24·71-s + 8·73-s − 16·77-s − 20·79-s − 4·83-s − 16·85-s − 4·89-s + 4·97-s + 4·101-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.51·7-s + 1.20·11-s + 0.970·17-s + 1.66·23-s + 4/5·25-s − 0.742·29-s − 2.15·31-s + 2.70·35-s − 1.31·37-s + 1.87·41-s + 1.21·43-s + 1.16·47-s − 0.549·53-s − 2.15·55-s − 1.04·59-s − 1.02·61-s + 1.95·67-s + 2.84·71-s + 0.936·73-s − 1.82·77-s − 2.25·79-s − 0.439·83-s − 1.73·85-s − 0.423·89-s + 0.406·97-s + 0.398·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21233664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 130 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 20 T + 240 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.88167858710803595727087085595, −7.82558676650077260235677786110, −7.29654440008988167172121262173, −7.22745404745803242089590525532, −6.76125500524777023911826131943, −6.43581499984762172646290301150, −5.95494801008077829095745797460, −5.67642853441712098474378789026, −5.02831703732541069481056573312, −4.92519029247021365464585717933, −3.98860244767094249152022917355, −3.87363686950431121127608338070, −3.55820281626850075455067063818, −3.55151630954207785930339717133, −2.60043140340997059868422639638, −2.53779806018088467459933808297, −1.29473671637629931787079199586, −1.17814215515877463189567270695, 0, 0,
1.17814215515877463189567270695, 1.29473671637629931787079199586, 2.53779806018088467459933808297, 2.60043140340997059868422639638, 3.55151630954207785930339717133, 3.55820281626850075455067063818, 3.87363686950431121127608338070, 3.98860244767094249152022917355, 4.92519029247021365464585717933, 5.02831703732541069481056573312, 5.67642853441712098474378789026, 5.95494801008077829095745797460, 6.43581499984762172646290301150, 6.76125500524777023911826131943, 7.22745404745803242089590525532, 7.29654440008988167172121262173, 7.82558676650077260235677786110, 7.88167858710803595727087085595
