| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s + 16-s − 18-s − 12·23-s + 24-s + 25-s − 27-s − 32-s + 36-s − 12·43-s + 12·46-s − 48-s − 2·49-s − 50-s − 12·53-s + 54-s + 64-s + 12·69-s − 72-s + 8·73-s − 75-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 − 2.50·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.176·32-s + 1/6·36-s − 1.82·43-s + 1.76·46-s − 0.144·48-s − 2/7·49-s − 0.141·50-s − 1.64·53-s + 0.136·54-s + 1/8·64-s + 1.44·69-s − 0.117·72-s + 0.936·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.617989795917646364552015710665, −8.955026259947280058225096969686, −8.320237225469500753093283173495, −8.028691934241130197940422293501, −7.52430148411020305138714516003, −6.85144109644362439697166318928, −6.34268542257277673475630397290, −6.00608229850580921448158933136, −5.28919876889298701185407020295, −4.69422935284846289307808963794, −3.94542085186066603008292301846, −3.29944944595194631605420364641, −2.26606102118794296161645987589, −1.52058294475544939746414628065, 0,
1.52058294475544939746414628065, 2.26606102118794296161645987589, 3.29944944595194631605420364641, 3.94542085186066603008292301846, 4.69422935284846289307808963794, 5.28919876889298701185407020295, 6.00608229850580921448158933136, 6.34268542257277673475630397290, 6.85144109644362439697166318928, 7.52430148411020305138714516003, 8.028691934241130197940422293501, 8.320237225469500753093283173495, 8.955026259947280058225096969686, 9.617989795917646364552015710665
