| L(s) = 1 | − 2-s + 3-s − 5-s − 6-s − 4·7-s + 8-s + 10-s + 5·11-s − 10·13-s + 4·14-s − 15-s − 16-s + 4·17-s + 7·19-s − 4·21-s − 5·22-s − 23-s + 24-s + 10·26-s − 27-s + 30-s + 2·31-s + 5·33-s − 4·34-s + 4·35-s − 37-s − 7·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 0.316·10-s + 1.50·11-s − 2.77·13-s + 1.06·14-s − 0.258·15-s − 1/4·16-s + 0.970·17-s + 1.60·19-s − 0.872·21-s − 1.06·22-s − 0.208·23-s + 0.204·24-s + 1.96·26-s − 0.192·27-s + 0.182·30-s + 0.359·31-s + 0.870·33-s − 0.685·34-s + 0.676·35-s − 0.164·37-s − 1.13·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7847917237\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7847917237\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 11 T + 74 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−12.62648675376300103373547009149, −12.14309058434703106274816141330, −11.86170958269909143406564398218, −11.14424201069254828347079896328, −10.41788120155298777952940914741, −9.864281256627199700013985873582, −9.588859492014329857682023800138, −9.249933012113699635041113890814, −9.031514446498088448474668699270, −7.993585237523661689107967609075, −7.55210403002109270418447210492, −7.20836337470889214761053674365, −6.83736112334465588892998076990, −5.78353454051427504962569689774, −5.46957335753293207276971995174, −4.22722423404120236916766006798, −4.02662175875836488805177239606, −2.90214945622279744677650796997, −2.58867224981759117577008902522, −0.856922317882564226388418665597,
0.856922317882564226388418665597, 2.58867224981759117577008902522, 2.90214945622279744677650796997, 4.02662175875836488805177239606, 4.22722423404120236916766006798, 5.46957335753293207276971995174, 5.78353454051427504962569689774, 6.83736112334465588892998076990, 7.20836337470889214761053674365, 7.55210403002109270418447210492, 7.993585237523661689107967609075, 9.031514446498088448474668699270, 9.249933012113699635041113890814, 9.588859492014329857682023800138, 9.864281256627199700013985873582, 10.41788120155298777952940914741, 11.14424201069254828347079896328, 11.86170958269909143406564398218, 12.14309058434703106274816141330, 12.62648675376300103373547009149
