| L(s) = 1 | − 8·9-s + 8·17-s − 8·49-s + 56·73-s + 30·81-s + 40·89-s − 8·97-s − 24·113-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 64·153-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 8/3·9-s + 1.94·17-s − 8/7·49-s + 6.55·73-s + 10/3·81-s + 4.23·89-s − 0.812·97-s − 2.25·113-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 − 5.17·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.53·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09557342255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09557342255\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 156 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.20695128834554289634957153088, −5.88236980264016694126443173766, −5.74222725580328759689472278247, −5.45650312689070597632997513622, −5.33395914366770940719920841716, −5.23239219339930398725845549850, −4.90746176483440821050628408819, −4.88075306982590602299189526919, −4.71325174597464760244445893366, −4.27951040639620182463719431911, −3.80024040507372117304280730239, −3.73628094036087465845492484338, −3.67589718369664024846053367661, −3.36315205204296972502696770368, −3.22446070395968310111151081924, −3.04242041541455659851972586012, −2.59934480451596507890230914292, −2.50382170594518980276275885311, −2.22044079258096950220665598539, −2.09798571444731714581454791888, −1.71643203010754154777401942547, −1.12022780370064467444368288640, −0.990269067382213135206580532005, −0.74964915937611576496034731022, −0.05205174942172795523837889809,
0.05205174942172795523837889809, 0.74964915937611576496034731022, 0.990269067382213135206580532005, 1.12022780370064467444368288640, 1.71643203010754154777401942547, 2.09798571444731714581454791888, 2.22044079258096950220665598539, 2.50382170594518980276275885311, 2.59934480451596507890230914292, 3.04242041541455659851972586012, 3.22446070395968310111151081924, 3.36315205204296972502696770368, 3.67589718369664024846053367661, 3.73628094036087465845492484338, 3.80024040507372117304280730239, 4.27951040639620182463719431911, 4.71325174597464760244445893366, 4.88075306982590602299189526919, 4.90746176483440821050628408819, 5.23239219339930398725845549850, 5.33395914366770940719920841716, 5.45650312689070597632997513622, 5.74222725580328759689472278247, 5.88236980264016694126443173766, 6.20695128834554289634957153088
