| L(s) = 1 | + 2·3-s + 5-s + 3·9-s + 11-s − 5·13-s + 2·15-s − 8·17-s + 5·19-s + 8·23-s + 5·25-s + 4·27-s − 3·29-s − 2·31-s + 2·33-s − 3·37-s − 10·39-s + 6·41-s + 7·43-s + 3·45-s − 12·47-s − 16·51-s − 11·53-s + 55-s + 10·57-s − 5·59-s − 20·61-s − 5·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 9-s + 0.301·11-s − 1.38·13-s + 0.516·15-s − 1.94·17-s + 1.14·19-s + 1.66·23-s + 25-s + 0.769·27-s − 0.557·29-s − 0.359·31-s + 0.348·33-s − 0.493·37-s − 1.60·39-s + 0.937·41-s + 1.06·43-s + 0.447·45-s − 1.75·47-s − 2.24·51-s − 1.51·53-s + 0.134·55-s + 1.32·57-s − 0.650·59-s − 2.56·61-s − 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88510464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.784548797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.784548797\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 3 T + 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 11 T + 122 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 132 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + T + 132 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 117 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 7 T + 164 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 25 T + 336 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
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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
−7.66328531982910988548536257136, −7.64992175954345097120988135535, −7.33126789711657316884512611365, −6.79917022602003021690101807611, −6.65380468460929706355078888468, −6.35837839237825744027627275949, −5.85410935915425778699537735219, −5.34953985618329060208957704906, −4.90378106355982411668098203641, −4.86044661479823303219741803988, −4.28217321843639640422893790064, −4.24471567835145342075536636457, −3.26464277823186490474947102453, −3.24488238917503924636695537243, −2.88047509841753621674424224774, −2.49571966549554907731089548591, −1.86465137484388749313564147516, −1.77154419026422537891331533282, −1.10682712970812488746740921956, −0.39781015255148828026636074956,
0.39781015255148828026636074956, 1.10682712970812488746740921956, 1.77154419026422537891331533282, 1.86465137484388749313564147516, 2.49571966549554907731089548591, 2.88047509841753621674424224774, 3.24488238917503924636695537243, 3.26464277823186490474947102453, 4.24471567835145342075536636457, 4.28217321843639640422893790064, 4.86044661479823303219741803988, 4.90378106355982411668098203641, 5.34953985618329060208957704906, 5.85410935915425778699537735219, 6.35837839237825744027627275949, 6.65380468460929706355078888468, 6.79917022602003021690101807611, 7.33126789711657316884512611365, 7.64992175954345097120988135535, 7.66328531982910988548536257136
