| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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_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$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 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$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 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$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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
−8.238652791846733237366421753761, −7.66908758047720391264226499729, −7.32086003888770472758561881109, −6.84226190389055380015636813097, −6.20216224223872773886523896480, −6.07004320452680737875400238025, −5.57359243140840511800252234526, −4.76904719725218092637602631779, −4.58490991266389509183701773720, −4.14819381609435149178453824071, −3.23095256917444535892471443877, −2.97751671234536144774660739957, −1.98228973052001758833718389919, −1.42403210907507863163255771797, 0,
1.42403210907507863163255771797, 1.98228973052001758833718389919, 2.97751671234536144774660739957, 3.23095256917444535892471443877, 4.14819381609435149178453824071, 4.58490991266389509183701773720, 4.76904719725218092637602631779, 5.57359243140840511800252234526, 6.07004320452680737875400238025, 6.20216224223872773886523896480, 6.84226190389055380015636813097, 7.32086003888770472758561881109, 7.66908758047720391264226499729, 8.238652791846733237366421753761
