| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 2·5-s + 4·6-s − 2·7-s + 4·8-s + 3·9-s − 4·10-s − 4·11-s + 6·12-s − 4·14-s − 4·15-s + 5·16-s + 4·17-s + 6·18-s − 6·20-s − 4·21-s − 8·22-s + 2·23-s + 8·24-s + 3·25-s + 4·27-s − 6·28-s + 6·29-s − 8·30-s + 6·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 1.26·10-s − 1.20·11-s + 1.73·12-s − 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.970·17-s + 1.41·18-s − 1.34·20-s − 0.872·21-s − 1.70·22-s + 0.417·23-s + 1.63·24-s + 3/5·25-s + 0.769·27-s − 1.13·28-s + 1.11·29-s − 1.46·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23328900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.66548449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.66548449\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 50 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 62 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 74 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 166 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 190 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 14 T + 162 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 170 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 162 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.104831685634010261759900015193, −8.098499103087583657475451098458, −7.66058854966668683307910019754, −7.48000178004827146794342295027, −6.99188271024768320241590856078, −6.68709558979684333487919915606, −6.24781483730177330978495430913, −5.90676265106400313178518159758, −5.41951119055425894776143952669, −5.07716608108932720102883246264, −4.63180378924614166120354546850, −4.30210584163713911751628972876, −3.86244569792114215727052485647, −3.60531673100145239796476888052, −3.04176620825163963617722866273, −2.88511413663854424887085834273, −2.32356403269450113003524184469, −2.25915821222134760717391553703, −0.900798939171146690098070981155, −0.888368752930052801004702887580,
0.888368752930052801004702887580, 0.900798939171146690098070981155, 2.25915821222134760717391553703, 2.32356403269450113003524184469, 2.88511413663854424887085834273, 3.04176620825163963617722866273, 3.60531673100145239796476888052, 3.86244569792114215727052485647, 4.30210584163713911751628972876, 4.63180378924614166120354546850, 5.07716608108932720102883246264, 5.41951119055425894776143952669, 5.90676265106400313178518159758, 6.24781483730177330978495430913, 6.68709558979684333487919915606, 6.99188271024768320241590856078, 7.48000178004827146794342295027, 7.66058854966668683307910019754, 8.098499103087583657475451098458, 8.104831685634010261759900015193
