| L(s) = 1 | + 6·4-s − 2·5-s − 3·9-s − 12·11-s + 19·16-s − 12·19-s − 12·20-s + 5·25-s − 4·29-s − 16·31-s − 18·36-s − 4·41-s − 72·44-s + 6·45-s − 2·49-s + 24·55-s − 16·59-s + 28·61-s + 36·64-s + 16·71-s − 72·76-s + 56·79-s − 38·80-s − 6·89-s + 24·95-s + 36·99-s + 30·100-s + ⋯ |
| L(s) = 1 | + 3·4-s − 0.894·5-s − 9-s − 3.61·11-s + 19/4·16-s − 2.75·19-s − 2.68·20-s + 25-s − 0.742·29-s − 2.87·31-s − 3·36-s − 0.624·41-s − 10.8·44-s + 0.894·45-s − 2/7·49-s + 3.23·55-s − 2.08·59-s + 3.58·61-s + 9/2·64-s + 1.89·71-s − 8.25·76-s + 6.30·79-s − 4.24·80-s − 0.635·89-s + 2.46·95-s + 3.61·99-s + 3·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9628350624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9628350624\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 10 T^{2} - 1269 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - 38 T^{2} - 1365 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 85 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 150 T^{2} + 15611 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.163469866028281169255793670101, −8.152309216388511979300039116679, −7.995865850317798147103130118861, −7.71623143776770987509794827095, −7.41742753396503173266614618721, −7.19330332384531695378771713515, −7.12801452526628138689754102129, −6.53827712553479990005753086006, −6.47784088149282347659265848763, −6.32566613803766210978412408224, −5.99898790889188780519397328454, −5.50209002240075575478134010520, −5.18890549455590836968474999796, −5.16772431360540474487607417530, −5.13920103323531374538216254764, −4.29632932896992717040354542330, −3.95470157567648578993265603997, −3.51415861428821133978789082534, −3.33336188981734894675419931831, −2.90851501749655848745796840625, −2.43958974720202154325717235390, −2.40383248464456408378155162931, −2.01086526851167627854172972250, −1.89691053359203334930380492168, −0.36062599657678425611783671956,
0.36062599657678425611783671956, 1.89691053359203334930380492168, 2.01086526851167627854172972250, 2.40383248464456408378155162931, 2.43958974720202154325717235390, 2.90851501749655848745796840625, 3.33336188981734894675419931831, 3.51415861428821133978789082534, 3.95470157567648578993265603997, 4.29632932896992717040354542330, 5.13920103323531374538216254764, 5.16772431360540474487607417530, 5.18890549455590836968474999796, 5.50209002240075575478134010520, 5.99898790889188780519397328454, 6.32566613803766210978412408224, 6.47784088149282347659265848763, 6.53827712553479990005753086006, 7.12801452526628138689754102129, 7.19330332384531695378771713515, 7.41742753396503173266614618721, 7.71623143776770987509794827095, 7.995865850317798147103130118861, 8.152309216388511979300039116679, 8.163469866028281169255793670101
