| L(s) = 1 | − 4·3-s − 4·5-s + 4·7-s + 10·9-s − 8·13-s + 16·15-s − 16·21-s − 20·27-s − 12·29-s + 12·31-s − 16·35-s − 16·37-s + 32·39-s − 40·45-s − 4·49-s − 20·53-s − 16·61-s + 40·63-s + 32·65-s + 16·67-s − 8·71-s − 8·73-s + 12·79-s + 35·81-s + 48·87-s − 8·89-s − 32·91-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.78·5-s + 1.51·7-s + 10/3·9-s − 2.21·13-s + 4.13·15-s − 3.49·21-s − 3.84·27-s − 2.22·29-s + 2.15·31-s − 2.70·35-s − 2.63·37-s + 5.12·39-s − 5.96·45-s − 4/7·49-s − 2.74·53-s − 2.04·61-s + 5.03·63-s + 3.96·65-s + 1.95·67-s − 0.949·71-s − 0.936·73-s + 1.35·79-s + 35/9·81-s + 5.14·87-s − 0.847·89-s − 3.35·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 16 T^{2} + 44 T^{3} + 118 T^{4} + 44 p T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 20 T^{2} - 60 T^{3} + 186 T^{4} - 60 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T^{2} - 32 T^{3} + 230 T^{4} - 32 p T^{5} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 56 T^{2} + 264 T^{3} + 1122 T^{4} + 264 p T^{5} + 56 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 36 T^{2} - 64 T^{3} + 662 T^{4} - 64 p T^{5} + 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 44 T^{2} - 64 T^{3} + 966 T^{4} - 64 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 144 T^{2} + 964 T^{3} + 6422 T^{4} + 964 p T^{5} + 144 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 1140 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 200 T^{2} + 1552 T^{3} + 11010 T^{4} + 1552 p T^{5} + 200 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 100 T^{2} + 192 T^{3} + 4726 T^{4} + 192 p T^{5} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 76 T^{2} - 256 T^{3} + 2726 T^{4} - 256 p T^{5} + 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T + 336 T^{2} + 3452 T^{3} + 30134 T^{4} + 3452 p T^{5} + 336 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 296 T^{2} + 2704 T^{3} + 27618 T^{4} + 2704 p T^{5} + 296 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 4 p T^{2} - 2960 T^{3} + 27190 T^{4} - 2960 p T^{5} + 4 p^{3} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 252 T^{2} + 1512 T^{3} + 25766 T^{4} + 1512 p T^{5} + 252 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 196 T^{2} + 1816 T^{3} + 18022 T^{4} + 1816 p T^{5} + 196 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 148 T^{2} + 44 T^{3} + 794 T^{4} + 44 p T^{5} + 148 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 116 T^{2} - 160 T^{3} + 5510 T^{4} - 160 p T^{5} + 116 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 156 T^{2} + 504 T^{3} + 10022 T^{4} + 504 p T^{5} + 156 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 768 p T^{5} + 164 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.59114718406316778683039080392, −6.26028391532533158243871709263, −6.03231731854026104154876893350, −6.00875653261882002845927317322, −5.89542822884915002247658874286, −5.25601866950955791846088824986, −5.23779967898676565664828235354, −5.12144938094795925397581555058, −5.11519260331973458721246701333, −4.74148713213143816435262184885, −4.55227221478712693960031752468, −4.45716802777566801135130492773, −4.40567691563585988635624126163, −3.78337667086529487549485514100, −3.70948368662147059097088814276, −3.68448347064441934312972789376, −3.45552041095801557250812552157, −2.91713018417804372195105863698, −2.68261032884944676707285555881, −2.32203425828271414153679950131, −2.23856332766896901428855623750, −1.64234283423106015247744593109, −1.59695698861670199209041097030, −1.20492633795042585721050146197, −1.14943319673942930481233004926, 0, 0, 0, 0,
1.14943319673942930481233004926, 1.20492633795042585721050146197, 1.59695698861670199209041097030, 1.64234283423106015247744593109, 2.23856332766896901428855623750, 2.32203425828271414153679950131, 2.68261032884944676707285555881, 2.91713018417804372195105863698, 3.45552041095801557250812552157, 3.68448347064441934312972789376, 3.70948368662147059097088814276, 3.78337667086529487549485514100, 4.40567691563585988635624126163, 4.45716802777566801135130492773, 4.55227221478712693960031752468, 4.74148713213143816435262184885, 5.11519260331973458721246701333, 5.12144938094795925397581555058, 5.23779967898676565664828235354, 5.25601866950955791846088824986, 5.89542822884915002247658874286, 6.00875653261882002845927317322, 6.03231731854026104154876893350, 6.26028391532533158243871709263, 6.59114718406316778683039080392
