| L(s) = 1 | − 2·2-s + 2·4-s − 6·5-s + 16·7-s + 12·10-s + 8·11-s − 6·13-s − 32·14-s − 4·16-s + 38·17-s − 12·20-s − 16·22-s − 40·23-s + 11·25-s + 12·26-s + 32·28-s − 88·31-s + 8·32-s − 76·34-s − 96·35-s − 6·37-s + 140·41-s + 72·43-s + 16·44-s + 80·46-s + 128·49-s − 22·50-s + ⋯ |
| L(s) = 1 | − 2-s + 1/2·4-s − 6/5·5-s + 16/7·7-s + 6/5·10-s + 8/11·11-s − 0.461·13-s − 2.28·14-s − 1/4·16-s + 2.23·17-s − 3/5·20-s − 0.727·22-s − 1.73·23-s + 0.439·25-s + 6/13·26-s + 8/7·28-s − 2.83·31-s + 1/4·32-s − 2.23·34-s − 2.74·35-s − 0.162·37-s + 3.41·41-s + 1.67·43-s + 4/11·44-s + 1.73·46-s + 2.61·49-s − 0.439·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9784123921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9784123921\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 6 T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 38 T + 722 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 658 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T + 800 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 238 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 44 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 72 T + 2592 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 56 T + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 1502 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 72 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 88 T + 3872 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 88 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 110 T + 6050 T^{2} - 110 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12338 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 48 T + 1152 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15166 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 114 T + 6498 T^{2} + 114 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−14.23602651858445477500073167637, −14.15886357832638319280918651839, −12.62924959685238902986063294303, −12.47636677792100653882590862659, −11.67597479320255105685364595619, −11.49901903319005434993482542731, −10.96430840121788125065986836627, −10.44556285674008257262123730504, −9.639434373450386469354397017371, −9.138106399083541669950966414132, −8.362558544002381284914929252314, −7.923884175503687284138153359726, −7.52441484965829760354957143396, −7.32213427269648630663092833128, −5.77026034946687879387660380905, −5.35773864414447254213450436352, −4.17151599413298949152826555678, −3.89414961913487455801271052203, −2.13577502529342843796375824345, −1.04637085856452170468959364729,
1.04637085856452170468959364729, 2.13577502529342843796375824345, 3.89414961913487455801271052203, 4.17151599413298949152826555678, 5.35773864414447254213450436352, 5.77026034946687879387660380905, 7.32213427269648630663092833128, 7.52441484965829760354957143396, 7.923884175503687284138153359726, 8.362558544002381284914929252314, 9.138106399083541669950966414132, 9.639434373450386469354397017371, 10.44556285674008257262123730504, 10.96430840121788125065986836627, 11.49901903319005434993482542731, 11.67597479320255105685364595619, 12.47636677792100653882590862659, 12.62924959685238902986063294303, 14.15886357832638319280918651839, 14.23602651858445477500073167637
