| L(s) = 1 | − 4·4-s + 10·16-s − 32·25-s + 32·37-s + 32·43-s − 20·64-s + 32·67-s + 128·100-s − 32·109-s + 24·121-s + 127-s + 131-s + 137-s + 139-s − 128·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 32·169-s − 128·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2·4-s + 5/2·16-s − 6.39·25-s + 5.26·37-s + 4.87·43-s − 5/2·64-s + 3.90·67-s + 64/5·100-s − 3.06·109-s + 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.5·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.46·169-s − 9.75·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.780194712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.780194712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 + 16 T^{2} + 112 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 + 16 T^{2} + 304 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 32 T^{2} + 672 T^{4} + 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 12 T^{2} + 246 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 20 T^{2} - 266 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T^{2} - 122 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 160 T^{2} + 9760 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 124 T^{2} + 7750 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 144 T^{2} + 9650 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 108 T^{2} + 7830 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 16 T^{2} + 3088 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 4 T + p T^{2} )^{8} \) |
| 71 | \( ( 1 - 92 T^{2} + 4006 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 192 T^{2} + 18624 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 12 T^{2} + 13302 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 12480 T^{4} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 288 T^{2} + 38304 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.37303672961170537431765995603, −4.11254192293616425546011532299, −4.07790657329125424346364828910, −4.04941692187784425798861801437, −3.80147030286710572017038446727, −3.76439805507459780394058443818, −3.74813508189320112399166107937, −3.70714456322951817413097422126, −3.36751742109271734002551314101, −3.26678328773502889005324196388, −2.93269313950883052376490100919, −2.67814893421447219806435719728, −2.54704989994720465817202363705, −2.45606857955471988267594075019, −2.44607790361874943455580051716, −2.32920166348671160027834771489, −2.29269338880787280863078958601, −1.69227093993554189138724634654, −1.59365687338599863805362656687, −1.43278217748465579084951167867, −1.42655888496382364617516425075, −0.828495447046292193856166088193, −0.62882234400709844725041021148, −0.56650184757782503892350269510, −0.36659667843215848779998801125,
0.36659667843215848779998801125, 0.56650184757782503892350269510, 0.62882234400709844725041021148, 0.828495447046292193856166088193, 1.42655888496382364617516425075, 1.43278217748465579084951167867, 1.59365687338599863805362656687, 1.69227093993554189138724634654, 2.29269338880787280863078958601, 2.32920166348671160027834771489, 2.44607790361874943455580051716, 2.45606857955471988267594075019, 2.54704989994720465817202363705, 2.67814893421447219806435719728, 2.93269313950883052376490100919, 3.26678328773502889005324196388, 3.36751742109271734002551314101, 3.70714456322951817413097422126, 3.74813508189320112399166107937, 3.76439805507459780394058443818, 3.80147030286710572017038446727, 4.04941692187784425798861801437, 4.07790657329125424346364828910, 4.11254192293616425546011532299, 4.37303672961170537431765995603
Plot not available for L-functions of degree greater than 10.
