| L(s) = 1 | + 6·9-s − 4·11-s − 2·25-s + 28·29-s − 40·71-s − 12·79-s + 9·81-s − 24·99-s + 12·109-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 46·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2·9-s − 1.20·11-s − 2/5·25-s + 5.19·29-s − 4.74·71-s − 1.35·79-s + 81-s − 2.41·99-s + 1.14·109-s − 3.09·121-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 + 3.53·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.305500023\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.305500023\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 53 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )^{2}( 1 + 19 T + p T^{2} )^{2} \) |
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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
−7.02429709409911982384316956666, −6.97698557531034818077968814631, −6.71213006440912943594182825039, −6.62158205562783857525839254366, −6.49841297282546506369378869593, −5.81398345663655089806799938639, −5.77165088977816873192129970274, −5.70311571367136857598680613055, −5.46956454392757222239026486666, −4.79381452988828101998530296666, −4.75634776953072748634218040842, −4.65492401667521883578483007137, −4.30203119775406946387832885858, −4.21177954198269661181060492219, −4.20130493206798685793526296188, −3.34906453776074021685784432036, −3.14892450851111631761051724016, −3.09910867758932953131867298308, −2.78712103641037275631822818127, −2.37650688826741982875389195836, −2.14361889691587553951370284885, −1.58661233479883260522956626911, −1.33550535515666630077581757046, −1.02963108604203505102248180128, −0.45562131095881848179281836825,
0.45562131095881848179281836825, 1.02963108604203505102248180128, 1.33550535515666630077581757046, 1.58661233479883260522956626911, 2.14361889691587553951370284885, 2.37650688826741982875389195836, 2.78712103641037275631822818127, 3.09910867758932953131867298308, 3.14892450851111631761051724016, 3.34906453776074021685784432036, 4.20130493206798685793526296188, 4.21177954198269661181060492219, 4.30203119775406946387832885858, 4.65492401667521883578483007137, 4.75634776953072748634218040842, 4.79381452988828101998530296666, 5.46956454392757222239026486666, 5.70311571367136857598680613055, 5.77165088977816873192129970274, 5.81398345663655089806799938639, 6.49841297282546506369378869593, 6.62158205562783857525839254366, 6.71213006440912943594182825039, 6.97698557531034818077968814631, 7.02429709409911982384316956666
