| L(s) = 1 | − 16·19-s − 4·25-s − 8·43-s − 2·49-s − 16·67-s + 40·73-s + 8·97-s + 40·121-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 + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 3.67·19-s − 4/5·25-s − 1.21·43-s − 2/7·49-s − 1.95·67-s + 4.68·73-s + 0.812·97-s + 3.63·121-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 + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.092776018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.092776018\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
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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
−5.94220038570519448194916089165, −5.84872182338975117187854715848, −5.45425466926994384859054781368, −5.43257936317300005251271268971, −5.31483672566720978174647816607, −4.91713313088572057797105714482, −4.59613838383280361134783584394, −4.52248975144401655074347346632, −4.43813017481226524427710331478, −4.22706873670196751718779378085, −3.97929265892836265502563388116, −3.80795997877267167287448626691, −3.54786167550386909182670310327, −3.15634647220214174491492430491, −3.07289500862368791382469936013, −3.02552300158137984709908375922, −2.56013378497228691770130318864, −2.11047854104509655797735025317, −1.99458599945328057610175429094, −1.99033217425363617417116270958, −1.77698228163816288316211788801, −1.47887309262312143815719468700, −0.62869112607706820213733435701, −0.53372856009719971710997387559, −0.49595681400993365209837501461,
0.49595681400993365209837501461, 0.53372856009719971710997387559, 0.62869112607706820213733435701, 1.47887309262312143815719468700, 1.77698228163816288316211788801, 1.99033217425363617417116270958, 1.99458599945328057610175429094, 2.11047854104509655797735025317, 2.56013378497228691770130318864, 3.02552300158137984709908375922, 3.07289500862368791382469936013, 3.15634647220214174491492430491, 3.54786167550386909182670310327, 3.80795997877267167287448626691, 3.97929265892836265502563388116, 4.22706873670196751718779378085, 4.43813017481226524427710331478, 4.52248975144401655074347346632, 4.59613838383280361134783584394, 4.91713313088572057797105714482, 5.31483672566720978174647816607, 5.43257936317300005251271268971, 5.45425466926994384859054781368, 5.84872182338975117187854715848, 5.94220038570519448194916089165
