| L(s) = 1 | + 4·3-s − 2·5-s − 4·7-s + 10·9-s + 2·13-s − 8·15-s − 10·17-s + 4·19-s − 16·21-s − 4·23-s − 8·25-s + 20·27-s + 11·29-s + 14·31-s + 8·35-s + 13·37-s + 8·39-s + 7·41-s + 12·43-s − 20·45-s + 15·47-s + 10·49-s − 40·51-s + 53-s + 16·57-s + 5·59-s + 3·61-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s − 1.51·7-s + 10/3·9-s + 0.554·13-s − 2.06·15-s − 2.42·17-s + 0.917·19-s − 3.49·21-s − 0.834·23-s − 8/5·25-s + 3.84·27-s + 2.04·29-s + 2.51·31-s + 1.35·35-s + 2.13·37-s + 1.28·39-s + 1.09·41-s + 1.82·43-s − 2.98·45-s + 2.18·47-s + 10/7·49-s − 5.60·51-s + 0.137·53-s + 2.11·57-s + 0.650·59-s + 0.384·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.68237006\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.68237006\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 12 T^{2} + 21 T^{3} + 68 T^{4} + 21 p T^{5} + 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 7 T^{2} + 232 T^{4} + 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 24 T^{2} - 49 T^{3} + 474 T^{4} - 49 p T^{5} + 24 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 69 T^{2} - 206 T^{3} + 1908 T^{4} - 206 p T^{5} + 69 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 11 T + 121 T^{2} - 718 T^{3} + 4782 T^{4} - 718 p T^{5} + 121 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 7 T + 52 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 13 T + 197 T^{2} - 1498 T^{3} + 11850 T^{4} - 1498 p T^{5} + 197 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 - 7 T + 125 T^{2} - 626 T^{3} + 6898 T^{4} - 626 p T^{5} + 125 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 12 T + 134 T^{2} - 659 T^{3} + 5190 T^{4} - 659 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 15 T + 232 T^{2} - 1923 T^{3} + 16614 T^{4} - 1923 p T^{5} + 232 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - T + 185 T^{2} - 201 T^{3} + 13976 T^{4} - 201 p T^{5} + 185 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 129 T^{2} - 435 T^{3} + 9596 T^{4} - 435 p T^{5} + 129 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 3 T + 17 T^{2} + 251 T^{3} + 2652 T^{4} + 251 p T^{5} + 17 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 257 T^{2} - 973 T^{3} + 25444 T^{4} - 973 p T^{5} + 257 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 5 T + 165 T^{2} - 1119 T^{3} + 14732 T^{4} - 1119 p T^{5} + 165 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 199 T^{2} + 652 T^{3} + 17348 T^{4} + 652 p T^{5} + 199 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 26 T + 317 T^{2} - 2554 T^{3} + 20452 T^{4} - 2554 p T^{5} + 317 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 171 T^{2} + 20 T^{3} + 15384 T^{4} + 20 p T^{5} + 171 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 - 7 T + 285 T^{2} - 1255 T^{3} + 34016 T^{4} - 1255 p T^{5} + 285 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 27 T + 491 T^{2} + 6242 T^{3} + 69216 T^{4} + 6242 p T^{5} + 491 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.18535617401852804132494813813, −5.81772792426996349962759484274, −5.58487452591324180050908019110, −5.51308456165119964214643463390, −5.27197914330156207007707302537, −4.61303771256068836046719912017, −4.48326714839826520408167253890, −4.47176411660843647095106733514, −4.46892459113199247742064672869, −3.97233242341582138660621618445, −3.90531205389734045700574054712, −3.89123347643662564648022710960, −3.66544532477085839255918278943, −3.09501277948135345890961308247, −2.96290861533994160386820347479, −2.88707682673903017447448911103, −2.82363234506714995320436458847, −2.40617814036859171240769685765, −2.11048085968526303500416918797, −2.10457102055332021738824245190, −1.96739873218145017615550098252, −1.12891854608539919332817774087, −0.864054895750017514213802042036, −0.810369323845452058477026214151, −0.44655101126165193144975689653,
0.44655101126165193144975689653, 0.810369323845452058477026214151, 0.864054895750017514213802042036, 1.12891854608539919332817774087, 1.96739873218145017615550098252, 2.10457102055332021738824245190, 2.11048085968526303500416918797, 2.40617814036859171240769685765, 2.82363234506714995320436458847, 2.88707682673903017447448911103, 2.96290861533994160386820347479, 3.09501277948135345890961308247, 3.66544532477085839255918278943, 3.89123347643662564648022710960, 3.90531205389734045700574054712, 3.97233242341582138660621618445, 4.46892459113199247742064672869, 4.47176411660843647095106733514, 4.48326714839826520408167253890, 4.61303771256068836046719912017, 5.27197914330156207007707302537, 5.51308456165119964214643463390, 5.58487452591324180050908019110, 5.81772792426996349962759484274, 6.18535617401852804132494813813
