| L(s) = 1 | − 4·2-s + 2·3-s + 12·4-s − 10·5-s − 8·6-s − 32·8-s − 5·9-s + 40·10-s − 68·11-s + 24·12-s + 52·13-s − 20·15-s + 80·16-s − 164·17-s + 20·18-s + 232·19-s − 120·20-s + 272·22-s − 198·23-s − 64·24-s + 75·25-s − 208·26-s + 26·27-s − 18·29-s + 80·30-s − 196·31-s − 192·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.384·3-s + 3/2·4-s − 0.894·5-s − 0.544·6-s − 1.41·8-s − 0.185·9-s + 1.26·10-s − 1.86·11-s + 0.577·12-s + 1.10·13-s − 0.344·15-s + 5/4·16-s − 2.33·17-s + 0.261·18-s + 2.80·19-s − 1.34·20-s + 2.63·22-s − 1.79·23-s − 0.544·24-s + 3/5·25-s − 1.56·26-s + 0.185·27-s − 0.115·29-s + 0.486·30-s − 1.13·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + p^{2} T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 68 T + 3634 T^{2} + 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 p T + 4334 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 164 T + 13606 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 232 T + 26438 T^{2} - 232 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 198 T + 23785 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 18 T + 33955 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 196 T + 27786 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 160 T + 101082 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 62 T + 72379 T^{2} + 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 198 T + 140065 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 164 T + 211426 T^{2} + 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 40 T + 5570 p T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 80 T - 85178 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 174 T + 105307 T^{2} - 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1054 T + 672761 T^{2} + 1054 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 832 T + 815278 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 820 T + 840150 T^{2} + 820 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 576 T + 880606 T^{2} + 576 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 298 T + 423841 T^{2} - 298 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 182 T + 1152523 T^{2} - 182 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 892 T + 1918278 T^{2} + 892 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.34776788013057545236386032035, −9.834004188612689274553040635034, −9.257107474720428561982165086115, −9.045259724049241437119985722194, −8.386456487273605606709753321467, −8.218072605488178108090091433399, −7.61973981069178400487584782718, −7.47298215952957045797716804917, −6.94995312692386906741403684678, −6.26890458702139957394183513160, −5.67672375789941247705982286114, −5.30683031437977801905639643482, −4.39236955133977824638796963731, −3.89902629033933013211748808005, −2.96210175062548845152882749686, −2.83608812274789522453266314827, −1.94779501270488225540316155032, −1.18544078469758491623642159273, 0, 0,
1.18544078469758491623642159273, 1.94779501270488225540316155032, 2.83608812274789522453266314827, 2.96210175062548845152882749686, 3.89902629033933013211748808005, 4.39236955133977824638796963731, 5.30683031437977801905639643482, 5.67672375789941247705982286114, 6.26890458702139957394183513160, 6.94995312692386906741403684678, 7.47298215952957045797716804917, 7.61973981069178400487584782718, 8.218072605488178108090091433399, 8.386456487273605606709753321467, 9.045259724049241437119985722194, 9.257107474720428561982165086115, 9.834004188612689274553040635034, 10.34776788013057545236386032035
