| L(s) = 1 | + 2·7-s − 9-s + 8·17-s + 12·23-s − 25-s + 16·31-s + 4·41-s − 12·47-s + 3·49-s − 2·63-s + 20·73-s − 16·79-s + 81-s + 28·89-s + 20·97-s − 24·103-s + 8·113-s + 16·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 8·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1/3·9-s + 1.94·17-s + 2.50·23-s − 1/5·25-s + 2.87·31-s + 0.624·41-s − 1.75·47-s + 3/7·49-s − 0.251·63-s + 2.34·73-s − 1.80·79-s + 1/9·81-s + 2.96·89-s + 2.03·97-s − 2.36·103-s + 0.752·113-s + 1.46·119-s + 2·121-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 − 0.646·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45158400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.187756023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.187756023\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.002075061002654404926659771185, −8.000673901920567185867698662362, −7.47460086003897666706180320874, −7.22451533687002391603383560624, −6.58025036144096006005323985714, −6.57741374316966919971178094462, −6.02719644976039541613614396718, −5.66829393349357877981500389951, −5.19357104995595522820114645151, −5.01545310312330946518257332653, −4.62957166393340043416945947317, −4.38355544833033116624190901370, −3.62100725764674165704067917935, −3.34424276831566552972489218175, −2.97869873185898411007797236530, −2.65202997368917322332952035425, −2.04763622866667309422715042517, −1.46694277412993968261493003627, −0.831322509849359062580693683731, −0.812368534521627519590523376819,
0.812368534521627519590523376819, 0.831322509849359062580693683731, 1.46694277412993968261493003627, 2.04763622866667309422715042517, 2.65202997368917322332952035425, 2.97869873185898411007797236530, 3.34424276831566552972489218175, 3.62100725764674165704067917935, 4.38355544833033116624190901370, 4.62957166393340043416945947317, 5.01545310312330946518257332653, 5.19357104995595522820114645151, 5.66829393349357877981500389951, 6.02719644976039541613614396718, 6.57741374316966919971178094462, 6.58025036144096006005323985714, 7.22451533687002391603383560624, 7.47460086003897666706180320874, 8.000673901920567185867698662362, 8.002075061002654404926659771185
