| L(s) = 1 | + 3-s − 5-s + 3·9-s − 2·11-s − 15-s + 4·17-s − 2·19-s − 23-s + 8·27-s + 18·29-s + 4·31-s − 2·33-s − 4·37-s − 2·41-s + 18·43-s − 3·45-s + 4·51-s + 10·53-s + 2·55-s − 2·57-s − 10·59-s + 9·61-s − 5·67-s − 69-s + 28·71-s + 12·73-s − 14·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 9-s − 0.603·11-s − 0.258·15-s + 0.970·17-s − 0.458·19-s − 0.208·23-s + 1.53·27-s + 3.34·29-s + 0.718·31-s − 0.348·33-s − 0.657·37-s − 0.312·41-s + 2.74·43-s − 0.447·45-s + 0.560·51-s + 1.37·53-s + 0.269·55-s − 0.264·57-s − 1.30·59-s + 1.15·61-s − 0.610·67-s − 0.120·69-s + 3.32·71-s + 1.40·73-s − 1.57·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3841600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.560697964\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.560697964\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 9 T + 20 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 71 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−9.333145953985018623932945278826, −8.895072215982615921592698034611, −8.435469862193168320642565655771, −8.255216967266607607130384055892, −7.961839202869787582449193212589, −7.45356241891997318824606331837, −6.99452400834371807650404981182, −6.81787296961755700746944720182, −6.24355323833479075269387145181, −5.86962907614305509419063657850, −5.22056502205986896003517885700, −4.88208856009997941801532672993, −4.31133469576105791455621490049, −4.18966732637487338207856671362, −3.50767898046996789985618916316, −2.93248917528691691862594305455, −2.64500267097379228229972168372, −2.10469204372960169934496241159, −1.10085507578061616991041183266, −0.808099252602085608671356573658,
0.808099252602085608671356573658, 1.10085507578061616991041183266, 2.10469204372960169934496241159, 2.64500267097379228229972168372, 2.93248917528691691862594305455, 3.50767898046996789985618916316, 4.18966732637487338207856671362, 4.31133469576105791455621490049, 4.88208856009997941801532672993, 5.22056502205986896003517885700, 5.86962907614305509419063657850, 6.24355323833479075269387145181, 6.81787296961755700746944720182, 6.99452400834371807650404981182, 7.45356241891997318824606331837, 7.961839202869787582449193212589, 8.255216967266607607130384055892, 8.435469862193168320642565655771, 8.895072215982615921592698034611, 9.333145953985018623932945278826
