| L(s) = 1 | − 2·9-s + 8·13-s + 12·17-s + 18·25-s + 18·49-s − 24·53-s − 15·81-s − 48·101-s − 24·113-s − 16·117-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 2.21·13-s + 2.91·17-s + 18/5·25-s + 18/7·49-s − 3.29·53-s − 5/3·81-s − 4.77·101-s − 2.25·113-s − 1.47·117-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.275294071\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.275294071\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 3 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 65 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.052819617094294380160781551358, −8.001500452478332955298605761066, −7.78432139796203097999391745948, −7.27268070954437664544399199482, −7.23434626904243823269085128066, −6.73888529220967120930714143886, −6.59999908147912859376864130892, −6.54904934030621723450562342239, −6.05229397049117612180975765342, −5.62967365083753600460579439536, −5.61202973468214076599314075972, −5.50724896718677580597447627175, −5.21898937662854559738389325959, −4.59100940752095021947675651536, −4.57791071373344686202891073234, −4.11256054277083003311805819273, −3.86899428235325612345899844575, −3.35523302780359735111402028897, −3.10231851921943527992165824507, −3.00929405415632434968429946521, −2.87182939557841694659686341312, −2.07637111531916380067842848935, −1.41446769528959812138942316526, −1.20280345372071301713816249436, −0.840251519037930368833641877088,
0.840251519037930368833641877088, 1.20280345372071301713816249436, 1.41446769528959812138942316526, 2.07637111531916380067842848935, 2.87182939557841694659686341312, 3.00929405415632434968429946521, 3.10231851921943527992165824507, 3.35523302780359735111402028897, 3.86899428235325612345899844575, 4.11256054277083003311805819273, 4.57791071373344686202891073234, 4.59100940752095021947675651536, 5.21898937662854559738389325959, 5.50724896718677580597447627175, 5.61202973468214076599314075972, 5.62967365083753600460579439536, 6.05229397049117612180975765342, 6.54904934030621723450562342239, 6.59999908147912859376864130892, 6.73888529220967120930714143886, 7.23434626904243823269085128066, 7.27268070954437664544399199482, 7.78432139796203097999391745948, 8.001500452478332955298605761066, 8.052819617094294380160781551358
