| L(s) = 1 | − 2-s + 4·5-s − 7-s + 8-s − 4·10-s − 5·13-s + 14-s − 16-s + 17-s − 6·19-s + 3·23-s + 2·25-s + 5·26-s + 6·29-s + 10·31-s − 34-s − 4·35-s + 2·37-s + 6·38-s + 4·40-s − 10·41-s − 3·43-s − 3·46-s − 12·47-s − 2·50-s − 14·53-s − 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.78·5-s − 0.377·7-s + 0.353·8-s − 1.26·10-s − 1.38·13-s + 0.267·14-s − 1/4·16-s + 0.242·17-s − 1.37·19-s + 0.625·23-s + 2/5·25-s + 0.980·26-s + 1.11·29-s + 1.79·31-s − 0.171·34-s − 0.676·35-s + 0.328·37-s + 0.973·38-s + 0.632·40-s − 1.56·41-s − 0.457·43-s − 0.442·46-s − 1.75·47-s − 0.282·50-s − 1.92·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2683044 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.311666488\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.311666488\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 3 T - 34 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 7 T - 10 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 11 T + 32 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 12 T + 47 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−9.684998437857251794941524981395, −9.409853787100275847070932061955, −8.898083995367773446485515643401, −8.311147222621082531374209485347, −8.161765406321850108179716757658, −7.78296940597206240876739587829, −7.12480110216719097874233586863, −6.54727868046055810501910210734, −6.37151689946421128543421996525, −6.30139576097670517632370992810, −5.40788974733831074366577549772, −5.15395565595146619610440614766, −4.57000675281170591265823023051, −4.48144974350261119265207534959, −3.33953354944953556862506048177, −3.08364651021702455422168352373, −2.31875389067506064379622677415, −2.01314488986879676842460818508, −1.47884643825066137927298182492, −0.48569593661245254563164000005,
0.48569593661245254563164000005, 1.47884643825066137927298182492, 2.01314488986879676842460818508, 2.31875389067506064379622677415, 3.08364651021702455422168352373, 3.33953354944953556862506048177, 4.48144974350261119265207534959, 4.57000675281170591265823023051, 5.15395565595146619610440614766, 5.40788974733831074366577549772, 6.30139576097670517632370992810, 6.37151689946421128543421996525, 6.54727868046055810501910210734, 7.12480110216719097874233586863, 7.78296940597206240876739587829, 8.161765406321850108179716757658, 8.311147222621082531374209485347, 8.898083995367773446485515643401, 9.409853787100275847070932061955, 9.684998437857251794941524981395
