| L(s) = 1 | + 8·9-s − 24·17-s − 32·41-s + 8·49-s + 24·73-s + 30·81-s − 24·89-s − 40·97-s + 40·113-s − 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 192·153-s + 157-s + 163-s + 167-s + 44·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 8/3·9-s − 5.82·17-s − 4.99·41-s + 8/7·49-s + 2.80·73-s + 10/3·81-s − 2.54·89-s − 4.06·97-s + 3.76·113-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 − 15.5·153-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7792892518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7792892518\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 84 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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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
−6.31192599994629443816179421541, −6.13750876012166040545444477804, −5.67807430364811971678562893232, −5.34875685075280458305508789334, −5.34011654922641314625779430203, −5.05221487771172827511651040678, −4.81829835325462410953757372426, −4.52408103662973909781699198621, −4.48101294386227015340897161105, −4.46536479420125400518651568974, −4.15549126709171501446856606202, −3.77263282140169461112918312702, −3.74615023843914494026088102957, −3.73860134702209565605510752218, −3.14214245085775495247024182010, −2.82336084462008636173033952304, −2.66687396725846354100161100346, −2.29601995891341981858499192254, −2.16724578394476069021648893376, −1.75155587832882000337387975984, −1.72950783519311556131506506940, −1.63171887486124918380389279639, −1.11046800699204804839025176366, −0.52910810802708064070589152590, −0.16232167247119800372126578819,
0.16232167247119800372126578819, 0.52910810802708064070589152590, 1.11046800699204804839025176366, 1.63171887486124918380389279639, 1.72950783519311556131506506940, 1.75155587832882000337387975984, 2.16724578394476069021648893376, 2.29601995891341981858499192254, 2.66687396725846354100161100346, 2.82336084462008636173033952304, 3.14214245085775495247024182010, 3.73860134702209565605510752218, 3.74615023843914494026088102957, 3.77263282140169461112918312702, 4.15549126709171501446856606202, 4.46536479420125400518651568974, 4.48101294386227015340897161105, 4.52408103662973909781699198621, 4.81829835325462410953757372426, 5.05221487771172827511651040678, 5.34011654922641314625779430203, 5.34875685075280458305508789334, 5.67807430364811971678562893232, 6.13750876012166040545444477804, 6.31192599994629443816179421541
