| L(s) = 1 | + 4-s + 2·5-s + 3·9-s + 4·19-s + 2·20-s + 5·25-s + 4·29-s + 20·31-s + 3·36-s + 12·41-s + 6·45-s − 4·59-s − 18·61-s − 64-s + 24·71-s + 4·76-s − 20·79-s + 9·81-s + 14·89-s + 8·95-s + 5·100-s − 30·101-s + 10·109-s + 4·116-s + 22·121-s + 20·124-s + 22·125-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 0.894·5-s + 9-s + 0.917·19-s + 0.447·20-s + 25-s + 0.742·29-s + 3.59·31-s + 1/2·36-s + 1.87·41-s + 0.894·45-s − 0.520·59-s − 2.30·61-s − 1/8·64-s + 2.84·71-s + 0.458·76-s − 2.25·79-s + 81-s + 1.48·89-s + 0.820·95-s + 1/2·100-s − 2.98·101-s + 0.957·109-s + 0.371·116-s + 2·121-s + 1.79·124-s + 1.96·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.972157557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.972157557\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 30 T^{2} - 1309 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 2 T - 55 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 9 T + 20 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−7.953842943244093041595531559095, −7.64559655388369570846117701483, −7.28215191276137394092523571041, −7.17895501754613818355807392718, −7.12258182110801217283522717325, −6.56490203397105135433768481944, −6.44921517520472811182374058573, −6.19577552967707319736947817390, −6.05766709280086773611765814882, −5.88428657233783682941557740431, −5.43560639884435732202053555320, −5.00118432280535199166807086174, −4.93192999322142060820396217217, −4.63395977760822236284572903696, −4.47060047473403568072882958158, −3.95106437028853899356553115478, −3.94005557007521707030496466276, −3.19718156470117231343074168530, −2.93846823101421540246862549715, −2.92051508017496846724223945429, −2.33423728391999526839941474582, −2.16947919147953350913857254703, −1.56559820086775594019784258294, −1.00916921836375522427923156611, −0.975520410458260499901207174180,
0.975520410458260499901207174180, 1.00916921836375522427923156611, 1.56559820086775594019784258294, 2.16947919147953350913857254703, 2.33423728391999526839941474582, 2.92051508017496846724223945429, 2.93846823101421540246862549715, 3.19718156470117231343074168530, 3.94005557007521707030496466276, 3.95106437028853899356553115478, 4.47060047473403568072882958158, 4.63395977760822236284572903696, 4.93192999322142060820396217217, 5.00118432280535199166807086174, 5.43560639884435732202053555320, 5.88428657233783682941557740431, 6.05766709280086773611765814882, 6.19577552967707319736947817390, 6.44921517520472811182374058573, 6.56490203397105135433768481944, 7.12258182110801217283522717325, 7.17895501754613818355807392718, 7.28215191276137394092523571041, 7.64559655388369570846117701483, 7.953842943244093041595531559095
