| L(s) = 1 | − 4·9-s + 24·29-s + 8·41-s + 12·49-s + 8·61-s − 6·81-s − 40·89-s + 8·101-s − 72·109-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 4/3·9-s + 4.45·29-s + 1.24·41-s + 12/7·49-s + 1.02·61-s − 2/3·81-s − 4.23·89-s + 0.796·101-s − 6.89·109-s + 1.81·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.38·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.322844536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.322844536\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 126 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.71109540722337397623949502591, −6.33355625110433723899208867520, −6.30101724688792077497253102667, −6.25463740027604736927391148023, −5.70964526337928929724202295719, −5.66172034698881502367642518451, −5.45831158967915928654549026155, −5.20237943943613302465092369871, −4.96131092250223794411676097617, −4.88432406759477666731122396895, −4.26878807028459189260222808150, −4.22054382921406884478613210866, −4.18290299958026347422169719205, −3.96981655492187148013825241809, −3.47472821642530814842251672059, −3.02084562593031995282203671071, −2.92797579620847002424235102555, −2.86217587259847144809412136301, −2.49053115599854101173423600993, −2.42379606216385048724638893119, −1.96187683237837979839434404500, −1.36485938206365033990291846106, −1.03745201013272084248129158929, −1.00577800743495396839109662713, −0.23393442398807279656439679550,
0.23393442398807279656439679550, 1.00577800743495396839109662713, 1.03745201013272084248129158929, 1.36485938206365033990291846106, 1.96187683237837979839434404500, 2.42379606216385048724638893119, 2.49053115599854101173423600993, 2.86217587259847144809412136301, 2.92797579620847002424235102555, 3.02084562593031995282203671071, 3.47472821642530814842251672059, 3.96981655492187148013825241809, 4.18290299958026347422169719205, 4.22054382921406884478613210866, 4.26878807028459189260222808150, 4.88432406759477666731122396895, 4.96131092250223794411676097617, 5.20237943943613302465092369871, 5.45831158967915928654549026155, 5.66172034698881502367642518451, 5.70964526337928929724202295719, 6.25463740027604736927391148023, 6.30101724688792077497253102667, 6.33355625110433723899208867520, 6.71109540722337397623949502591
