| L(s) = 1 | − 4·4-s − 16·5-s + 12·16-s + 64·20-s − 40·23-s + 66·25-s + 24·31-s − 156·37-s − 136·47-s + 52·49-s + 28·53-s + 160·59-s − 32·64-s − 88·67-s + 48·71-s − 192·80-s + 164·89-s + 160·92-s + 192·97-s − 264·100-s + 368·103-s − 96·113-s + 640·115-s − 96·124-s + 576·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 4-s − 3.19·5-s + 3/4·16-s + 16/5·20-s − 1.73·23-s + 2.63·25-s + 0.774·31-s − 4.21·37-s − 2.89·47-s + 1.06·49-s + 0.528·53-s + 2.71·59-s − 1/2·64-s − 1.31·67-s + 0.676·71-s − 2.39·80-s + 1.84·89-s + 1.73·92-s + 1.97·97-s − 2.63·100-s + 3.57·103-s − 0.849·113-s + 5.56·115-s − 0.774·124-s + 4.60·125-s + 0.00787·127-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5330593979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5330593979\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 + 8 T + 63 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 432 T^{2} + 91103 T^{4} - 432 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 192 T^{2} - 18817 T^{4} - 192 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 372 T^{2} + 246086 T^{4} - 372 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 20 T + 570 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3216 T^{2} + 3996959 T^{4} - 3216 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 506 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 78 T + 4211 T^{2} + 78 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 5632 T^{2} + 13386303 T^{4} - 5632 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6052 T^{2} + 15821478 T^{4} - 6052 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 68 T + 2874 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 14 T + 5619 T^{2} - 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 - 80 T + 4674 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12352 T^{2} + 64479906 T^{4} - 12352 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 44 T + 4170 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 24 T + 9794 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 5664 T^{2} + 40756034 T^{4} - 5664 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 3828 T^{2} + 44392358 T^{4} - 3828 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 8820 T^{2} + 75829574 T^{4} - 8820 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 82 T + 15795 T^{2} - 82 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 96 T + 14495 T^{2} - 96 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.47373473112412291512202855313, −5.88117892806595600022253567921, −5.62517096784143529467541619533, −5.59826968910123528829703949030, −5.57577068972738668464258655344, −5.03819409381054259427792360580, −4.87103470096309322567485001051, −4.75729878016932172055497827922, −4.35362446953968933844372130300, −4.31113042872712152527064087352, −4.04044731303883603665222816241, −3.89787050509909333481468467426, −3.76837052861940342831979873704, −3.41918972764297770028405074819, −3.22992720737991951876927992424, −3.20959469696355533757025299908, −3.02269123424991878083092983327, −2.18519747206300243646550699326, −1.96339243850693384987085683993, −1.91043340332515319003112028818, −1.74412699344691529288053079964, −0.919150591307922173234516973400, −0.63959282604494376621994223806, −0.39622517167594997374098449404, −0.23326665489649943033503477798,
0.23326665489649943033503477798, 0.39622517167594997374098449404, 0.63959282604494376621994223806, 0.919150591307922173234516973400, 1.74412699344691529288053079964, 1.91043340332515319003112028818, 1.96339243850693384987085683993, 2.18519747206300243646550699326, 3.02269123424991878083092983327, 3.20959469696355533757025299908, 3.22992720737991951876927992424, 3.41918972764297770028405074819, 3.76837052861940342831979873704, 3.89787050509909333481468467426, 4.04044731303883603665222816241, 4.31113042872712152527064087352, 4.35362446953968933844372130300, 4.75729878016932172055497827922, 4.87103470096309322567485001051, 5.03819409381054259427792360580, 5.57577068972738668464258655344, 5.59826968910123528829703949030, 5.62517096784143529467541619533, 5.88117892806595600022253567921, 6.47373473112412291512202855313
