| L(s) = 1 | + 2·2-s + 3·4-s − 4·5-s + 4·8-s + 6·9-s − 8·10-s + 2·11-s + 12·13-s + 5·16-s + 12·18-s + 6·19-s − 12·20-s + 4·22-s − 8·23-s + 2·25-s + 24·26-s − 12·29-s + 6·32-s + 18·36-s + 12·38-s − 16·40-s + 6·44-s − 24·45-s − 16·46-s − 16·47-s + 4·49-s + 4·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 1.78·5-s + 1.41·8-s + 2·9-s − 2.52·10-s + 0.603·11-s + 3.32·13-s + 5/4·16-s + 2.82·18-s + 1.37·19-s − 2.68·20-s + 0.852·22-s − 1.66·23-s + 2/5·25-s + 4.70·26-s − 2.22·29-s + 1.06·32-s + 3·36-s + 1.94·38-s − 2.52·40-s + 0.904·44-s − 3.57·45-s − 2.35·46-s − 2.33·47-s + 4/7·49-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.102951284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.102951284\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 112 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
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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
−11.40677190889836730843795386311, −11.37021291886839740926569813547, −10.72523298992940951435099704187, −10.38531235028644058462722378859, −9.522834874879285750360637390848, −9.391912512251361860167234710530, −8.329802527891076939901352356975, −8.079253126539737526014767719253, −7.68870893120935972763862995258, −7.16032036554465826996825384644, −6.74256334850287325158031929717, −6.09433694221273873810179442384, −5.82065799858462033371346802944, −5.00995859071554686710487126919, −4.15142222813522390336190755924, −3.99897741002942507583190063143, −3.63695398521856232345341333850, −3.43237194673800342557675305555, −1.75216151321566150103481050222, −1.32344574004910212184053116959,
1.32344574004910212184053116959, 1.75216151321566150103481050222, 3.43237194673800342557675305555, 3.63695398521856232345341333850, 3.99897741002942507583190063143, 4.15142222813522390336190755924, 5.00995859071554686710487126919, 5.82065799858462033371346802944, 6.09433694221273873810179442384, 6.74256334850287325158031929717, 7.16032036554465826996825384644, 7.68870893120935972763862995258, 8.079253126539737526014767719253, 8.329802527891076939901352356975, 9.391912512251361860167234710530, 9.522834874879285750360637390848, 10.38531235028644058462722378859, 10.72523298992940951435099704187, 11.37021291886839740926569813547, 11.40677190889836730843795386311
