| L(s) = 1 | − 2·2-s + 3·3-s + 3·4-s − 6·6-s + 2·7-s − 4·8-s + 6·9-s + 9·12-s − 4·14-s + 5·16-s − 12·18-s − 8·19-s + 6·21-s − 12·24-s − 2·25-s + 9·27-s + 6·28-s − 18·29-s − 6·32-s + 18·36-s + 16·38-s − 12·42-s + 4·43-s + 15·48-s − 11·49-s + 4·50-s + 18·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 3/2·4-s − 2.44·6-s + 0.755·7-s − 1.41·8-s + 2·9-s + 2.59·12-s − 1.06·14-s + 5/4·16-s − 2.82·18-s − 1.83·19-s + 1.30·21-s − 2.44·24-s − 2/5·25-s + 1.73·27-s + 1.13·28-s − 3.34·29-s − 1.06·32-s + 3·36-s + 2.59·38-s − 1.85·42-s + 0.609·43-s + 2.16·48-s − 1.57·49-s + 0.565·50-s + 2.47·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.014876791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014876791\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−13.94405735128223976422455890320, −13.41443002050458089445793186250, −12.66236941589837179367898930518, −12.57579759618386388604901078259, −11.39982173113675917207642660543, −11.15459925579274283049481845686, −10.58443456365459907580287938768, −9.930035272462788786741653902004, −9.314696248225270502049248431688, −9.155950241897008087526607507583, −8.307453311780821382709945834705, −8.212552959238264029829531990035, −7.54327311687729818305855281059, −7.10187701398086923190915939762, −6.29901505145278870809570277864, −5.35039909813264016226809681016, −4.13413131291149390841434116951, −3.54725453331496676552181256947, −2.24970756960208927516047071219, −1.89648213860845155691770044950,
1.89648213860845155691770044950, 2.24970756960208927516047071219, 3.54725453331496676552181256947, 4.13413131291149390841434116951, 5.35039909813264016226809681016, 6.29901505145278870809570277864, 7.10187701398086923190915939762, 7.54327311687729818305855281059, 8.212552959238264029829531990035, 8.307453311780821382709945834705, 9.155950241897008087526607507583, 9.314696248225270502049248431688, 9.930035272462788786741653902004, 10.58443456365459907580287938768, 11.15459925579274283049481845686, 11.39982173113675917207642660543, 12.57579759618386388604901078259, 12.66236941589837179367898930518, 13.41443002050458089445793186250, 13.94405735128223976422455890320
