| L(s) = 1 | + 6·3-s − 2·7-s + 27·9-s + 74·11-s + 98·13-s − 78·17-s − 80·19-s − 12·21-s + 40·23-s + 108·27-s + 50·29-s − 12·31-s + 444·33-s − 34·37-s + 588·39-s + 344·41-s + 216·43-s + 876·47-s + 478·49-s − 468·51-s − 634·53-s − 480·57-s + 666·59-s + 244·61-s − 54·63-s − 980·67-s + 240·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.107·7-s + 9-s + 2.02·11-s + 2.09·13-s − 1.11·17-s − 0.965·19-s − 0.124·21-s + 0.362·23-s + 0.769·27-s + 0.320·29-s − 0.0695·31-s + 2.34·33-s − 0.151·37-s + 2.41·39-s + 1.31·41-s + 0.766·43-s + 2.71·47-s + 1.39·49-s − 1.28·51-s − 1.64·53-s − 1.11·57-s + 1.46·59-s + 0.512·61-s − 0.107·63-s − 1.78·67-s + 0.418·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.032068507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.032068507\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( 1 + 2 T - 474 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 74 T + 3902 T^{2} - 74 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 98 T + 6666 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 78 T + 8122 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 80 T + 7062 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 40 T + 16478 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 50 T + 43082 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 17822 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 34 T + 20970 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 344 T + 162782 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 876 T + 374206 T^{2} - 876 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 634 T + 398114 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 666 T + 211918 T^{2} - 666 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 p T + 336750 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 980 T + 692502 T^{2} + 980 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 308 T + 553262 T^{2} - 308 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1412 T + 1090194 T^{2} + 1412 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1052 T + 1237470 T^{2} - 1052 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 248 T + 1125926 T^{2} - 248 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 684 T + 1229686 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1840 T + 2539650 T^{2} + 1840 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.43669562557646710216880811053, −10.09245799164554459136708975461, −9.144174632756916275628198108521, −9.007976491055808414617689994244, −8.945264403197844514153305746816, −8.604158444337866192502816749637, −7.82508521488061386966684652062, −7.49158388172521085927568310919, −6.73167191810326642606979893971, −6.59977515281625863186987996291, −6.02489169493280744427615631852, −5.69008575347177264158215849218, −4.39103178831439506798597622572, −4.36659344771878606735265885512, −3.79091446818388749661376483537, −3.45150280540685777266176179308, −2.57134343184453226264169563816, −2.05391789906745569314344577312, −1.25673155729579359690571827219, −0.827335399725095797662936038173,
0.827335399725095797662936038173, 1.25673155729579359690571827219, 2.05391789906745569314344577312, 2.57134343184453226264169563816, 3.45150280540685777266176179308, 3.79091446818388749661376483537, 4.36659344771878606735265885512, 4.39103178831439506798597622572, 5.69008575347177264158215849218, 6.02489169493280744427615631852, 6.59977515281625863186987996291, 6.73167191810326642606979893971, 7.49158388172521085927568310919, 7.82508521488061386966684652062, 8.604158444337866192502816749637, 8.945264403197844514153305746816, 9.007976491055808414617689994244, 9.144174632756916275628198108521, 10.09245799164554459136708975461, 10.43669562557646710216880811053
