| L(s) = 1 | + 2·2-s + 2·4-s + 3·5-s + 4·8-s + 3·9-s + 6·10-s + 6·11-s − 2·13-s + 8·16-s − 4·17-s + 6·18-s − 5·19-s + 6·20-s + 12·22-s − 3·23-s + 5·25-s − 4·26-s − 10·29-s + 3·31-s + 8·32-s − 8·34-s + 6·36-s + 4·37-s − 10·38-s + 12·40-s − 12·41-s − 2·43-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.34·5-s + 1.41·8-s + 9-s + 1.89·10-s + 1.80·11-s − 0.554·13-s + 2·16-s − 0.970·17-s + 1.41·18-s − 1.14·19-s + 1.34·20-s + 2.55·22-s − 0.625·23-s + 25-s − 0.784·26-s − 1.85·29-s + 0.538·31-s + 1.41·32-s − 1.37·34-s + 36-s + 0.657·37-s − 1.62·38-s + 1.89·40-s − 1.87·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.509418347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.509418347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 6 T - 31 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 13 T + 96 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 3 T - 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.76930336283261111601906239084, −10.23108550068423304512400264697, −10.14210756010956150683716927382, −9.613822800953605735632754125364, −9.068090885679127142698830178716, −8.860086291941955628914762568412, −8.029994355851387376684716450214, −7.58478912016162589526448266690, −6.93909622768209686013821525932, −6.53987412117126131891073002678, −6.43600706997210459217068424160, −5.74503378456894661603662577883, −5.25236592027102490946166241079, −4.60456407011118138898494534144, −4.47414788048595035254945532500, −3.74241046224243632189636717717, −3.52097569193699672231567763651, −2.09201118382306068447341750076, −2.08139855061317724919995613635, −1.33070982566271400609252630348,
1.33070982566271400609252630348, 2.08139855061317724919995613635, 2.09201118382306068447341750076, 3.52097569193699672231567763651, 3.74241046224243632189636717717, 4.47414788048595035254945532500, 4.60456407011118138898494534144, 5.25236592027102490946166241079, 5.74503378456894661603662577883, 6.43600706997210459217068424160, 6.53987412117126131891073002678, 6.93909622768209686013821525932, 7.58478912016162589526448266690, 8.029994355851387376684716450214, 8.860086291941955628914762568412, 9.068090885679127142698830178716, 9.613822800953605735632754125364, 10.14210756010956150683716927382, 10.23108550068423304512400264697, 10.76930336283261111601906239084
