| L(s) = 1 | − 2·2-s + 4-s − 2·5-s + 2·7-s + 9-s + 4·10-s − 4·11-s − 4·14-s + 16-s − 4·17-s − 2·18-s − 2·20-s + 8·22-s − 4·23-s − 25-s + 2·28-s + 4·29-s − 6·31-s + 2·32-s + 8·34-s − 4·35-s + 36-s − 9·37-s + 4·41-s − 2·43-s − 4·44-s − 2·45-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1/2·4-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.26·10-s − 1.20·11-s − 1.06·14-s + 1/4·16-s − 0.970·17-s − 0.471·18-s − 0.447·20-s + 1.70·22-s − 0.834·23-s − 1/5·25-s + 0.377·28-s + 0.742·29-s − 1.07·31-s + 0.353·32-s + 1.37·34-s − 0.676·35-s + 1/6·36-s − 1.47·37-s + 0.624·41-s − 0.304·43-s − 0.603·44-s − 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12321 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 37 | $C_2$ | \( 1 + 9 T + p T^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + p T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 25 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 27 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 55 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 33 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 163 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 157 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 9 T + 108 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
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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
−16.5271758839, −16.1252484985, −15.7239371941, −15.3209409891, −14.8157424512, −14.2353122508, −13.5195027680, −13.2085340329, −12.4176114847, −11.9883705585, −11.4486825413, −10.8721192969, −10.3661568404, −10.0303539329, −9.15824043023, −8.81874401037, −8.23669537503, −7.78062878351, −7.44986888206, −6.57938913582, −5.69877306269, −4.81997696255, −4.26861691674, −3.17417661452, −1.88995258507, 0,
1.88995258507, 3.17417661452, 4.26861691674, 4.81997696255, 5.69877306269, 6.57938913582, 7.44986888206, 7.78062878351, 8.23669537503, 8.81874401037, 9.15824043023, 10.0303539329, 10.3661568404, 10.8721192969, 11.4486825413, 11.9883705585, 12.4176114847, 13.2085340329, 13.5195027680, 14.2353122508, 14.8157424512, 15.3209409891, 15.7239371941, 16.1252484985, 16.5271758839
