| L(s) = 1 | − 4·4-s − 4·9-s + 8·11-s + 4·16-s + 24·23-s + 8·25-s + 16·36-s − 48·43-s − 32·44-s − 24·53-s + 16·64-s − 48·67-s + 48·79-s + 14·81-s − 96·92-s − 32·99-s − 32·100-s − 48·109-s − 64·113-s + 68·121-s + 127-s + 131-s + 137-s + 139-s − 16·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 2·4-s − 4/3·9-s + 2.41·11-s + 16-s + 5.00·23-s + 8/5·25-s + 8/3·36-s − 7.31·43-s − 4.82·44-s − 3.29·53-s + 2·64-s − 5.86·67-s + 5.40·79-s + 14/9·81-s − 10.0·92-s − 3.21·99-s − 3.19·100-s − 4.59·109-s − 6.02·113-s + 6.18·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.525559991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.525559991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 4 T^{2} + 2 T^{4} - 16 T^{6} - 29 T^{8} - 16 p^{2} T^{10} + 2 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 - 8 T^{2} + 16 T^{4} + 16 T^{6} + 79 T^{8} + 16 p^{2} T^{10} + 16 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 13 | \( ( 1 - 8 T^{2} + 336 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 + 48 T^{2} + 1248 T^{4} + 22944 T^{6} + 365759 T^{8} + 22944 p^{2} T^{10} + 1248 p^{4} T^{12} + 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 19 | \( 1 + 36 T^{2} + 258 T^{4} + 11376 T^{6} + 479267 T^{8} + 11376 p^{2} T^{10} + 258 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 12 T + 82 T^{2} - 408 T^{3} + 1731 T^{4} - 408 p T^{5} + 82 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 8 T^{2} + 546 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 - 76 T^{2} + 2698 T^{4} - 87856 T^{6} + 2999827 T^{8} - 87856 p^{2} T^{10} + 2698 p^{4} T^{12} - 76 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 + 68 T^{2} + 3255 T^{4} + 68 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 144 T^{2} + 8448 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( 1 + 4 T^{2} + 202 T^{4} - 18416 T^{6} - 4880429 T^{8} - 18416 p^{2} T^{10} + 202 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( 1 + 228 T^{2} + 32034 T^{4} + 2961264 T^{6} + 204928835 T^{8} + 2961264 p^{2} T^{10} + 32034 p^{4} T^{12} + 228 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 136 T^{2} + 7888 T^{4} - 430576 T^{6} + 29860207 T^{8} - 430576 p^{2} T^{10} + 7888 p^{4} T^{12} - 136 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - p T^{2} )^{8} \) |
| 73 | \( 1 + 240 T^{2} + 33120 T^{4} + 3317280 T^{6} + 263921759 T^{8} + 3317280 p^{2} T^{10} + 33120 p^{4} T^{12} + 240 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 - 24 T + 374 T^{2} - 4368 T^{3} + 42051 T^{4} - 4368 p T^{5} + 374 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 228 T^{2} + 26382 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 96 T^{2} - 7872 T^{4} + 1344 p T^{6} + 19727 p^{2} T^{8} + 1344 p^{3} T^{10} - 7872 p^{4} T^{12} + 96 p^{6} T^{14} + p^{8} T^{16} \) |
| 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
−5.13885795630265937721859480868, −4.73846363882754766337796961162, −4.67674669675786952208463342722, −4.65486833401976767578009016432, −4.61163912412591879440921713705, −4.46296294112764369034375767041, −4.20776111107111844015763204837, −3.89240958156620867380961484308, −3.78324925026089151581192836263, −3.75472251604220480279438898430, −3.42252921196814146072245511860, −3.33173709225538002804303779675, −3.14035176331194501768954292421, −3.07058775315708753709272116459, −2.92062664197518786800661790288, −2.90908270449910520266204144177, −2.66886328101910351944450285616, −2.30341511599795589282870391887, −1.76009892583259562989946231930, −1.63764318702709672350522757386, −1.54996099286322208854724778594, −1.51158937446589444491072604622, −1.08049430412254144553839463690, −0.51352975735602340360992150858, −0.45902328050941650411881204748,
0.45902328050941650411881204748, 0.51352975735602340360992150858, 1.08049430412254144553839463690, 1.51158937446589444491072604622, 1.54996099286322208854724778594, 1.63764318702709672350522757386, 1.76009892583259562989946231930, 2.30341511599795589282870391887, 2.66886328101910351944450285616, 2.90908270449910520266204144177, 2.92062664197518786800661790288, 3.07058775315708753709272116459, 3.14035176331194501768954292421, 3.33173709225538002804303779675, 3.42252921196814146072245511860, 3.75472251604220480279438898430, 3.78324925026089151581192836263, 3.89240958156620867380961484308, 4.20776111107111844015763204837, 4.46296294112764369034375767041, 4.61163912412591879440921713705, 4.65486833401976767578009016432, 4.67674669675786952208463342722, 4.73846363882754766337796961162, 5.13885795630265937721859480868
Plot not available for L-functions of degree greater than 10.
