| L(s) = 1 | + 4-s + 4·5-s − 6·9-s − 6·11-s − 10·19-s + 4·20-s + 5·25-s + 16·29-s + 4·31-s − 6·36-s + 12·41-s − 6·44-s − 24·45-s − 2·49-s − 24·55-s − 8·59-s − 12·61-s − 64-s − 24·71-s − 10·76-s + 28·79-s + 9·81-s + 4·89-s − 40·95-s + 36·99-s + 5·100-s + 4·109-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s − 2·9-s − 1.80·11-s − 2.29·19-s + 0.894·20-s + 25-s + 2.97·29-s + 0.718·31-s − 36-s + 1.87·41-s − 0.904·44-s − 3.57·45-s − 2/7·49-s − 3.23·55-s − 1.04·59-s − 1.53·61-s − 1/8·64-s − 2.84·71-s − 1.14·76-s + 3.15·79-s + 81-s + 0.423·89-s − 4.10·95-s + 3.61·99-s + 1/2·100-s + 0.383·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8037536565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8037536565\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - 14 T + 117 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.00288960029022323344523879439, −10.36885151170820898870890211826, −10.33937039601442105215947901050, −10.25321854300404979655528034078, −10.15172993684019108314376753627, −9.257965064754940141499293867865, −9.130416190934001159258178329616, −8.806108920344191086922236346895, −8.732000280773706090483137127007, −7.963653473364360370948674694096, −7.903820222778613211112110032568, −7.897795448188747920263290077983, −7.08691500602834560351759685181, −6.42033470391414466067033464248, −6.37698392419049473313456733974, −6.18661944655002757176010237946, −5.86277139977059120087424334488, −5.31991896541835435014788028969, −5.17048229971270943599787253056, −4.48972306656910139104697797137, −4.32092512076614076650127781573, −3.04827988601823267908083379659, −2.84781537432164549982239978598, −2.50548909992439682869609315126, −1.98793399665740071309754400575,
1.98793399665740071309754400575, 2.50548909992439682869609315126, 2.84781537432164549982239978598, 3.04827988601823267908083379659, 4.32092512076614076650127781573, 4.48972306656910139104697797137, 5.17048229971270943599787253056, 5.31991896541835435014788028969, 5.86277139977059120087424334488, 6.18661944655002757176010237946, 6.37698392419049473313456733974, 6.42033470391414466067033464248, 7.08691500602834560351759685181, 7.897795448188747920263290077983, 7.903820222778613211112110032568, 7.963653473364360370948674694096, 8.732000280773706090483137127007, 8.806108920344191086922236346895, 9.130416190934001159258178329616, 9.257965064754940141499293867865, 10.15172993684019108314376753627, 10.25321854300404979655528034078, 10.33937039601442105215947901050, 10.36885151170820898870890211826, 11.00288960029022323344523879439
