| L(s) = 1 | − 2·5-s + 3·7-s + 6·11-s + 3·13-s − 4·17-s + 6·19-s + 6·23-s + 5·25-s + 8·29-s − 6·35-s + 14·37-s − 8·41-s − 12·43-s + 6·47-s + 7·49-s + 8·53-s − 12·55-s + 6·59-s + 61-s − 6·65-s − 3·67-s − 24·71-s − 30·73-s + 18·77-s + 9·79-s − 12·83-s + 8·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.13·7-s + 1.80·11-s + 0.832·13-s − 0.970·17-s + 1.37·19-s + 1.25·23-s + 25-s + 1.48·29-s − 1.01·35-s + 2.30·37-s − 1.24·41-s − 1.82·43-s + 0.875·47-s + 49-s + 1.09·53-s − 1.61·55-s + 0.781·59-s + 0.128·61-s − 0.744·65-s − 0.366·67-s − 2.84·71-s − 3.51·73-s + 2.05·77-s + 1.01·79-s − 1.31·83-s + 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6718464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.579713803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.579713803\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 8 T + 23 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 12 T + 101 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 9 T + 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−8.891667103155716382087337568851, −8.655604439665148787117898009333, −8.500865303779837356527538259968, −8.028887110238818119419288197843, −7.36844600912264891053400855178, −7.21219828811602097362528959425, −6.96634118827827064162849464350, −6.37336752372904864037437477029, −6.09741577203547591242424465180, −5.55730025082324528431218020971, −5.07066646026319733799880558189, −4.45837459255601040058079262012, −4.43729049925276825025392590420, −4.05159505793556041382238428323, −3.30102469354888504938231680871, −3.08153343481860954322300187124, −2.49525301079519857485053380653, −1.41979953575957238649747144745, −1.38565204603463967075572696870, −0.72387144762728601049807936173,
0.72387144762728601049807936173, 1.38565204603463967075572696870, 1.41979953575957238649747144745, 2.49525301079519857485053380653, 3.08153343481860954322300187124, 3.30102469354888504938231680871, 4.05159505793556041382238428323, 4.43729049925276825025392590420, 4.45837459255601040058079262012, 5.07066646026319733799880558189, 5.55730025082324528431218020971, 6.09741577203547591242424465180, 6.37336752372904864037437477029, 6.96634118827827064162849464350, 7.21219828811602097362528959425, 7.36844600912264891053400855178, 8.028887110238818119419288197843, 8.500865303779837356527538259968, 8.655604439665148787117898009333, 8.891667103155716382087337568851
