| L(s) = 1 | − 4-s + 2·5-s + 4·11-s + 16-s + 4·19-s − 2·20-s − 25-s − 12·29-s + 12·31-s − 12·41-s − 4·44-s − 49-s + 8·55-s − 16·59-s + 20·61-s − 64-s + 28·71-s − 4·76-s − 8·79-s + 2·80-s + 20·89-s + 8·95-s + 100-s + 20·101-s + 12·109-s + 12·116-s − 10·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s + 1.20·11-s + 1/4·16-s + 0.917·19-s − 0.447·20-s − 1/5·25-s − 2.22·29-s + 2.15·31-s − 1.87·41-s − 0.603·44-s − 1/7·49-s + 1.07·55-s − 2.08·59-s + 2.56·61-s − 1/8·64-s + 3.32·71-s − 0.458·76-s − 0.900·79-s + 0.223·80-s + 2.11·89-s + 0.820·95-s + 1/10·100-s + 1.99·101-s + 1.14·109-s + 1.11·116-s − 0.909·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.118743739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.118743739\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−10.74423386917910053261749379624, −10.11390651499434925418445489318, −9.949401906157975379336025401173, −9.520520359786936001837909184030, −9.146495760488311143127034002750, −8.841540788479496506418318143254, −8.107620373490479955672406980675, −7.926107802115565613681613333342, −7.21119637536265728910856406109, −6.67291692828615502546291759996, −6.39715154140196961986798094406, −5.77585435797111961215641278645, −5.38839019017613361890560745660, −4.86017933399314605250365526660, −4.33070952328280847170156845796, −3.49040150299360966035809250033, −3.42869511443778481440939090764, −2.26787165417138267047105949320, −1.74385858215399389597469014219, −0.854284030856195579160614206449,
0.854284030856195579160614206449, 1.74385858215399389597469014219, 2.26787165417138267047105949320, 3.42869511443778481440939090764, 3.49040150299360966035809250033, 4.33070952328280847170156845796, 4.86017933399314605250365526660, 5.38839019017613361890560745660, 5.77585435797111961215641278645, 6.39715154140196961986798094406, 6.67291692828615502546291759996, 7.21119637536265728910856406109, 7.926107802115565613681613333342, 8.107620373490479955672406980675, 8.841540788479496506418318143254, 9.146495760488311143127034002750, 9.520520359786936001837909184030, 9.949401906157975379336025401173, 10.11390651499434925418445489318, 10.74423386917910053261749379624
