L(s) = 1 | − 2·5-s + 4·11-s + 3·13-s + 14·17-s − 10·19-s + 4·23-s + 5·25-s − 29-s − 3·31-s + 22·37-s + 9·41-s − 5·43-s − 3·47-s − 6·53-s − 8·55-s + 7·59-s + 3·61-s − 6·65-s − 13·67-s + 16·71-s − 14·73-s + 9·79-s − 83-s − 28·85-s + 30·89-s + 20·95-s − 17·97-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 1.20·11-s + 0.832·13-s + 3.39·17-s − 2.29·19-s + 0.834·23-s + 25-s − 0.185·29-s − 0.538·31-s + 3.61·37-s + 1.40·41-s − 0.762·43-s − 0.437·47-s − 0.824·53-s − 1.07·55-s + 0.911·59-s + 0.384·61-s − 0.744·65-s − 1.58·67-s + 1.89·71-s − 1.63·73-s + 1.01·79-s − 0.109·83-s − 3.03·85-s + 3.17·89-s + 2.05·95-s − 1.72·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28005264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.933243138\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.933243138\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 3 T - 38 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + T - 82 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−8.187828525982981068499444327334, −7.931676395544953236966674735474, −7.78526603135592780587181944990, −7.45060997636354154487219080316, −6.97903817468824076086733755140, −6.41719282161575892942509193761, −6.17598240454344442940570901985, −6.11962553044319164611175347136, −5.44851101758007809096921599802, −5.15809943197861207258692524444, −4.47513793999166938105946293531, −4.35102022701928291115922571930, −3.74681453293789987868940252277, −3.71593576267856666396638783750, −3.02783595385104151264205309093, −2.87851373190549796172974051975, −2.07020576387531820079886064849, −1.50200045185499493019281639102, −0.858519095714874694257790457167, −0.75528049050771825133950652097,
0.75528049050771825133950652097, 0.858519095714874694257790457167, 1.50200045185499493019281639102, 2.07020576387531820079886064849, 2.87851373190549796172974051975, 3.02783595385104151264205309093, 3.71593576267856666396638783750, 3.74681453293789987868940252277, 4.35102022701928291115922571930, 4.47513793999166938105946293531, 5.15809943197861207258692524444, 5.44851101758007809096921599802, 6.11962553044319164611175347136, 6.17598240454344442940570901985, 6.41719282161575892942509193761, 6.97903817468824076086733755140, 7.45060997636354154487219080316, 7.78526603135592780587181944990, 7.931676395544953236966674735474, 8.187828525982981068499444327334