| L(s) = 1 | + 3-s + 9-s + 2·13-s + 6·25-s + 27-s + 2·39-s + 8·43-s − 14·49-s + 4·61-s + 6·75-s + 16·79-s + 81-s + 32·103-s + 2·117-s + 6·121-s + 127-s + 8·129-s + 131-s + 137-s + 139-s − 14·147-s + 149-s + 151-s + 157-s + 163-s + 167-s − 9·169-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.554·13-s + 6/5·25-s + 0.192·27-s + 0.320·39-s + 1.21·43-s − 2·49-s + 0.512·61-s + 0.692·75-s + 1.80·79-s + 1/9·81-s + 3.15·103-s + 0.184·117-s + 6/11·121-s + 0.0887·127-s + 0.704·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.15·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.692·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 292032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 292032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.328485625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.328485625\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.772689809590030263545165501480, −8.502604378015558457427421797267, −7.901441127993153509867944962933, −7.54746892931376049163450890695, −7.01732573053818861025815429439, −6.43477943177296487064051522724, −6.15664609264897815114302053968, −5.41381246649210083038571077824, −4.85380136920362411144637893615, −4.43323672227839498383660878681, −3.65386553034271336103677246074, −3.28035223298828958205046645456, −2.56902804312253384791144002212, −1.85189123120923416965849656530, −0.915311920404725519256267968532,
0.915311920404725519256267968532, 1.85189123120923416965849656530, 2.56902804312253384791144002212, 3.28035223298828958205046645456, 3.65386553034271336103677246074, 4.43323672227839498383660878681, 4.85380136920362411144637893615, 5.41381246649210083038571077824, 6.15664609264897815114302053968, 6.43477943177296487064051522724, 7.01732573053818861025815429439, 7.54746892931376049163450890695, 7.901441127993153509867944962933, 8.502604378015558457427421797267, 8.772689809590030263545165501480
