| L(s) = 1 | + 4-s + 4·5-s − 4·11-s + 4·20-s + 5·25-s + 16·31-s − 4·41-s − 4·44-s − 10·49-s − 16·55-s + 20·59-s − 4·61-s − 64-s + 48·71-s + 40·89-s + 5·100-s + 16·101-s − 40·109-s + 26·121-s + 16·124-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s − 1.20·11-s + 0.894·20-s + 25-s + 2.87·31-s − 0.624·41-s − 0.603·44-s − 1.42·49-s − 2.15·55-s + 2.60·59-s − 0.512·61-s − 1/8·64-s + 5.69·71-s + 4.23·89-s + 1/2·100-s + 1.59·101-s − 3.83·109-s + 2.36·121-s + 1.43·124-s − 0.357·125-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.845571561\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.845571561\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 + 30 T^{2} + 371 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 70 T^{2} + 3051 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 70 T^{2} + 411 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 150 T^{2} + 15611 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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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
−7.33666129176799793675013286716, −6.94242673823540578553136253178, −6.89981233279765125201687627531, −6.54772183649938157911162299386, −6.50214875311581585307363303897, −6.09245236927974655739083009946, −6.06096053503610792140618667317, −5.95173467864363145707395547817, −5.31998575769499049995256797310, −5.22822187667719928742493290796, −5.06433871285258856648427331210, −4.99256660246070083994003521149, −4.62670169705209815385098383564, −4.28469006205999367807306801180, −3.91573912978990611958041656851, −3.55550428933728432772196903035, −3.29609372545562301919233635451, −3.18271539405182893107735811771, −2.48227465803464685672432730936, −2.34799070293649203093814176380, −2.24935550316776924702207370883, −2.14009939371197766997804279024, −1.39407201238702464335676466093, −1.07704626064651636921553692714, −0.56578773997756096362111234065,
0.56578773997756096362111234065, 1.07704626064651636921553692714, 1.39407201238702464335676466093, 2.14009939371197766997804279024, 2.24935550316776924702207370883, 2.34799070293649203093814176380, 2.48227465803464685672432730936, 3.18271539405182893107735811771, 3.29609372545562301919233635451, 3.55550428933728432772196903035, 3.91573912978990611958041656851, 4.28469006205999367807306801180, 4.62670169705209815385098383564, 4.99256660246070083994003521149, 5.06433871285258856648427331210, 5.22822187667719928742493290796, 5.31998575769499049995256797310, 5.95173467864363145707395547817, 6.06096053503610792140618667317, 6.09245236927974655739083009946, 6.50214875311581585307363303897, 6.54772183649938157911162299386, 6.89981233279765125201687627531, 6.94242673823540578553136253178, 7.33666129176799793675013286716
