| L(s) = 1 | − 3-s − 4-s + 4·7-s − 2·9-s + 12-s + 8·13-s + 16-s − 6·19-s − 4·21-s − 4·25-s + 5·27-s − 4·28-s − 6·31-s + 2·36-s − 9·37-s − 8·39-s − 2·43-s − 48-s + 6·49-s − 8·52-s + 6·57-s + 14·61-s − 8·63-s − 64-s − 14·67-s − 17·73-s + 4·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 1.51·7-s − 2/3·9-s + 0.288·12-s + 2.21·13-s + 1/4·16-s − 1.37·19-s − 0.872·21-s − 4/5·25-s + 0.962·27-s − 0.755·28-s − 1.07·31-s + 1/3·36-s − 1.47·37-s − 1.28·39-s − 0.304·43-s − 0.144·48-s + 6/7·49-s − 1.10·52-s + 0.794·57-s + 1.79·61-s − 1.00·63-s − 1/8·64-s − 1.71·67-s − 1.98·73-s + 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7130201573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7130201573\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 3 T + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 5 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 47 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 112 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−12.17730161422958432786250970440, −11.45519903646355040652703683355, −11.28222094403448255206001825356, −10.65076289774599749787834007442, −10.25179142665266532494560919399, −8.977652295765378241283525733900, −8.636800990183078728823040941053, −8.312258680421605704040467597161, −7.47571868393168734332950009233, −6.42878923810303542043677298614, −5.84203656771386670626861194420, −5.21930288652091318854550860391, −4.35140731607977908867926467980, −3.55934054749465102130917053231, −1.73729238958677251693021747431,
1.73729238958677251693021747431, 3.55934054749465102130917053231, 4.35140731607977908867926467980, 5.21930288652091318854550860391, 5.84203656771386670626861194420, 6.42878923810303542043677298614, 7.47571868393168734332950009233, 8.312258680421605704040467597161, 8.636800990183078728823040941053, 8.977652295765378241283525733900, 10.25179142665266532494560919399, 10.65076289774599749787834007442, 11.28222094403448255206001825356, 11.45519903646355040652703683355, 12.17730161422958432786250970440
