| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·10-s − 4·11-s + 5·16-s + 4·17-s − 10·19-s + 6·20-s + 8·22-s + 2·23-s − 25-s + 4·29-s − 12·31-s − 6·32-s − 8·34-s + 4·37-s + 20·38-s − 8·40-s + 4·43-s − 12·44-s − 4·46-s + 2·50-s + 12·53-s − 8·55-s − 8·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.41·8-s − 1.26·10-s − 1.20·11-s + 5/4·16-s + 0.970·17-s − 2.29·19-s + 1.34·20-s + 1.70·22-s + 0.417·23-s − 1/5·25-s + 0.742·29-s − 2.15·31-s − 1.06·32-s − 1.37·34-s + 0.657·37-s + 3.24·38-s − 1.26·40-s + 0.609·43-s − 1.80·44-s − 0.589·46-s + 0.282·50-s + 1.64·53-s − 1.07·55-s − 1.05·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 3 p T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 118 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 143 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 143 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 174 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.54971107766653956070480491456, −7.51520901043969511407660084003, −7.21533349882079743176154591595, −6.65029855042273025578529124128, −6.23311800672535171381941753116, −6.12406417185450335289190515018, −5.58670672241762335966478457075, −5.54802319154076693826343105147, −4.92210242290622280731260502350, −4.55550698621807828327975736456, −3.99145639330950275952539782242, −3.70816674682945070201229758295, −2.92447412002799934950487610594, −2.74983060598549025951020948510, −2.40150637121654632253891681281, −1.82171791204364726840246623034, −1.59004598603295821190051766644, −1.00607691179480837755584785816, 0, 0,
1.00607691179480837755584785816, 1.59004598603295821190051766644, 1.82171791204364726840246623034, 2.40150637121654632253891681281, 2.74983060598549025951020948510, 2.92447412002799934950487610594, 3.70816674682945070201229758295, 3.99145639330950275952539782242, 4.55550698621807828327975736456, 4.92210242290622280731260502350, 5.54802319154076693826343105147, 5.58670672241762335966478457075, 6.12406417185450335289190515018, 6.23311800672535171381941753116, 6.65029855042273025578529124128, 7.21533349882079743176154591595, 7.51520901043969511407660084003, 7.54971107766653956070480491456
