| L(s) = 1 | − 4·3-s + 4·5-s + 4·7-s + 8·9-s + 2·13-s − 16·15-s − 10·17-s − 8·19-s − 16·21-s + 4·23-s + 11·25-s − 12·27-s + 16·35-s − 2·37-s − 8·39-s + 12·43-s + 32·45-s − 4·47-s + 8·49-s + 40·51-s + 14·53-s + 32·57-s − 8·59-s + 8·61-s + 32·63-s + 8·65-s + 20·67-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.78·5-s + 1.51·7-s + 8/3·9-s + 0.554·13-s − 4.13·15-s − 2.42·17-s − 1.83·19-s − 3.49·21-s + 0.834·23-s + 11/5·25-s − 2.30·27-s + 2.70·35-s − 0.328·37-s − 1.28·39-s + 1.82·43-s + 4.77·45-s − 0.583·47-s + 8/7·49-s + 5.60·51-s + 1.92·53-s + 4.23·57-s − 1.04·59-s + 1.02·61-s + 4.03·63-s + 0.992·65-s + 2.44·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.039637411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039637411\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−11.65259582350454511875998243899, −11.17537145608867586345175392932, −10.86126265487268488627904125241, −10.76419019559640746208847008356, −10.49161607431376272760035512623, −9.497375091892541281233676851995, −9.261371095628691839644381630784, −8.531854203808900664650264952255, −8.310363181742190250595575054547, −7.20767670096183399373053095905, −6.63658903681055329547280022023, −6.47976866008577400329105873132, −5.94493156480446112546251983154, −5.44125418197532702960391007894, −5.04727876082139398117591533819, −4.58041243288640931632434359529, −4.05574374549856144630671210715, −2.14675616168741467365200642355, −2.13202886647570508909171568080, −0.873977880245997273451414525901,
0.873977880245997273451414525901, 2.13202886647570508909171568080, 2.14675616168741467365200642355, 4.05574374549856144630671210715, 4.58041243288640931632434359529, 5.04727876082139398117591533819, 5.44125418197532702960391007894, 5.94493156480446112546251983154, 6.47976866008577400329105873132, 6.63658903681055329547280022023, 7.20767670096183399373053095905, 8.310363181742190250595575054547, 8.531854203808900664650264952255, 9.261371095628691839644381630784, 9.497375091892541281233676851995, 10.49161607431376272760035512623, 10.76419019559640746208847008356, 10.86126265487268488627904125241, 11.17537145608867586345175392932, 11.65259582350454511875998243899
