| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s + 16-s − 18-s − 6·19-s − 6·23-s + 24-s − 4·25-s − 27-s + 29-s − 32-s + 36-s + 6·38-s + 12·43-s + 6·46-s + 12·47-s − 48-s − 9·49-s + 4·50-s + 15·53-s + 54-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 1/4·16-s − 0.235·18-s − 1.37·19-s − 1.25·23-s + 0.204·24-s − 4/5·25-s − 0.192·27-s + 0.185·29-s − 0.176·32-s + 1/6·36-s + 0.973·38-s + 1.82·43-s + 0.884·46-s + 1.75·47-s − 0.144·48-s − 9/7·49-s + 0.565·50-s + 2.06·53-s + 0.136·54-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7256149140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7256149140\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 60 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 37 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 100 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−8.626793986932346547464817598600, −7.964461269282043577667728901264, −7.71560017656923596923423134083, −7.17336110101153378790356120495, −6.75422494377097661106355094883, −6.12174770253650357687831547524, −5.89043542421135098754617568087, −5.49008518986183088665699033495, −4.63003468909153503465783637310, −4.14329102010991170229013578606, −3.83154497536137796918451104161, −2.80801704952176125883657953356, −2.25393030169539466834734492218, −1.60435875669211744858163023275, −0.51827519228656907828416160782,
0.51827519228656907828416160782, 1.60435875669211744858163023275, 2.25393030169539466834734492218, 2.80801704952176125883657953356, 3.83154497536137796918451104161, 4.14329102010991170229013578606, 4.63003468909153503465783637310, 5.49008518986183088665699033495, 5.89043542421135098754617568087, 6.12174770253650357687831547524, 6.75422494377097661106355094883, 7.17336110101153378790356120495, 7.71560017656923596923423134083, 7.964461269282043577667728901264, 8.626793986932346547464817598600
