| L(s) = 1 | − 2·9-s − 10·11-s − 8·16-s + 8·19-s + 12·29-s + 22·31-s + 6·41-s + 2·49-s + 26·59-s + 4·61-s − 6·71-s + 6·79-s + 9·81-s + 44·89-s + 20·99-s − 40·101-s + 20·109-s + 47·121-s + 127-s + 131-s + 137-s + 139-s + 16·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 2/3·9-s − 3.01·11-s − 2·16-s + 1.83·19-s + 2.22·29-s + 3.95·31-s + 0.937·41-s + 2/7·49-s + 3.38·59-s + 0.512·61-s − 0.712·71-s + 0.675·79-s + 81-s + 4.66·89-s + 2.01·99-s − 3.98·101-s + 1.91·109-s + 4.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.030618944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.030618944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2^3$ | \( 1 + 2 T^{2} - 5 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 11 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2^3$ | \( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 13 T + 110 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^3$ | \( 1 + 130 T^{2} + 11571 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 150 T^{2} + 15611 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 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
−7.42584389064162749856250396401, −7.41385766365285020383087581793, −6.68165226746623280108307325033, −6.65683350308739395586190870447, −6.62089289914645223108860893076, −6.40123561983874708979956670106, −6.05091010935331164249618248812, −5.58247572620362713017075825332, −5.42283779657018683830100475560, −5.38089018669705810499644459563, −4.87973779683630144735189094226, −4.87928862053081705539908590467, −4.73078216039212121863003844393, −4.33739966935243122411489819397, −4.08917449835051142183666907042, −3.68681719864304278934129306662, −3.23207463059311447882211134460, −3.01391217317471544344551824862, −2.74354540061040157233017072113, −2.43064897782382035766196632386, −2.39805812712337016090306291765, −2.23574114640935197687191516558, −1.21644565261061426013391711182, −0.836096114131515295864866120225, −0.49182336451834100820117362578,
0.49182336451834100820117362578, 0.836096114131515295864866120225, 1.21644565261061426013391711182, 2.23574114640935197687191516558, 2.39805812712337016090306291765, 2.43064897782382035766196632386, 2.74354540061040157233017072113, 3.01391217317471544344551824862, 3.23207463059311447882211134460, 3.68681719864304278934129306662, 4.08917449835051142183666907042, 4.33739966935243122411489819397, 4.73078216039212121863003844393, 4.87928862053081705539908590467, 4.87973779683630144735189094226, 5.38089018669705810499644459563, 5.42283779657018683830100475560, 5.58247572620362713017075825332, 6.05091010935331164249618248812, 6.40123561983874708979956670106, 6.62089289914645223108860893076, 6.65683350308739395586190870447, 6.68165226746623280108307325033, 7.41385766365285020383087581793, 7.42584389064162749856250396401
