| L(s) = 1 | − 2·2-s + 3-s + 4-s − 2·6-s + 10·7-s + 2·8-s + 3·9-s − 3·11-s + 12-s − 14·13-s − 20·14-s − 4·16-s + 3·17-s − 6·18-s − 19-s + 10·21-s + 6·22-s − 9·23-s + 2·24-s + 13·25-s + 28·26-s + 8·27-s + 10·28-s − 3·29-s − 4·31-s + 2·32-s − 3·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 1/2·4-s − 0.816·6-s + 3.77·7-s + 0.707·8-s + 9-s − 0.904·11-s + 0.288·12-s − 3.88·13-s − 5.34·14-s − 16-s + 0.727·17-s − 1.41·18-s − 0.229·19-s + 2.18·21-s + 1.27·22-s − 1.87·23-s + 0.408·24-s + 13/5·25-s + 5.49·26-s + 1.53·27-s + 1.88·28-s − 0.557·29-s − 0.718·31-s + 0.353·32-s − 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.259678676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259678676\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 13 T^{2} + 84 T^{4} - 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 3 T - 19 T^{2} + 18 T^{3} + 342 T^{4} + 18 p T^{5} - 19 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + T - 29 T^{2} - 8 T^{3} + 520 T^{4} - 8 p T^{5} - 29 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 9 T + 77 T^{2} + 450 T^{3} + 2592 T^{4} + 450 p T^{5} + 77 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3 T + 59 T^{2} + 168 T^{3} + 2382 T^{4} + 168 p T^{5} + 59 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 6 T - 10 T^{2} - 132 T^{3} - 441 T^{4} - 132 p T^{5} - 10 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 27 T + 383 T^{2} + 3780 T^{3} + 27882 T^{4} + 3780 p T^{5} + 383 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 169 T^{2} + 11484 T^{4} - 169 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - T^{2} - 4488 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 27 T + 419 T^{2} + 4752 T^{3} + 41832 T^{4} + 4752 p T^{5} + 419 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 50 T^{2} - 228 T^{3} - 2241 T^{4} - 228 p T^{5} + 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 15 T + 101 T^{2} + 270 T^{3} - 5640 T^{4} + 270 p T^{5} + 101 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 10 T + 138 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 25 T + 306 T^{2} - 25 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 304 T^{2} + 36750 T^{4} - 304 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 3 T + 47 T^{2} - 132 T^{3} - 5718 T^{4} - 132 p T^{5} + 47 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 10 T - 86 T^{2} - 80 T^{3} + 15487 T^{4} - 80 p T^{5} - 86 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.77209431337103679357293691406, −7.63170759648148811334073149278, −7.61141488750175650607224273534, −7.48500656783013238600624416809, −6.92655921548222456488448209823, −6.82551194206921806856566443833, −6.61957965092883382274136243099, −6.32493837977638099870206769462, −5.53646853375080175619108399257, −5.32258941404889102390313224497, −5.03586786604847722971286312233, −4.98211054160208841873850350271, −4.94688324918873929578179666240, −4.84064375399941326872485327503, −4.41035211781027377003162373816, −3.92063457148674118812806012423, −3.89656772066875050484194647667, −3.18597415995688318318606313287, −2.77675224190490280796416456709, −2.53328435539838703189232745358, −1.99450657905141344499325334863, −1.84218974492706792076454564884, −1.79035854669682259963359720508, −1.18091875699200575784955799890, −0.44578731320238724266552267906,
0.44578731320238724266552267906, 1.18091875699200575784955799890, 1.79035854669682259963359720508, 1.84218974492706792076454564884, 1.99450657905141344499325334863, 2.53328435539838703189232745358, 2.77675224190490280796416456709, 3.18597415995688318318606313287, 3.89656772066875050484194647667, 3.92063457148674118812806012423, 4.41035211781027377003162373816, 4.84064375399941326872485327503, 4.94688324918873929578179666240, 4.98211054160208841873850350271, 5.03586786604847722971286312233, 5.32258941404889102390313224497, 5.53646853375080175619108399257, 6.32493837977638099870206769462, 6.61957965092883382274136243099, 6.82551194206921806856566443833, 6.92655921548222456488448209823, 7.48500656783013238600624416809, 7.61141488750175650607224273534, 7.63170759648148811334073149278, 7.77209431337103679357293691406
