| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 7-s − 4·8-s + 3·9-s − 4·10-s + 3·11-s − 6·12-s − 3·13-s + 2·14-s − 4·15-s + 5·16-s + 3·17-s − 6·18-s + 3·19-s + 6·20-s + 2·21-s − 6·22-s + 23-s + 8·24-s + 3·25-s + 6·26-s − 4·27-s − 3·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.377·7-s − 1.41·8-s + 9-s − 1.26·10-s + 0.904·11-s − 1.73·12-s − 0.832·13-s + 0.534·14-s − 1.03·15-s + 5/4·16-s + 0.727·17-s − 1.41·18-s + 0.688·19-s + 1.34·20-s + 0.436·21-s − 1.27·22-s + 0.208·23-s + 1.63·24-s + 3/5·25-s + 1.17·26-s − 0.769·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9878845338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9878845338\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T + 38 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 104 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 18 T + 166 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 158 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 92 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 7 T + 170 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 21 T + 280 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.949991477259471093589151146249, −9.785166473869818352386352040370, −9.233962082308994120919370484186, −9.132533672252278719758903061035, −8.362721335182838771891454882484, −8.166292991905687590382797175022, −7.37109103034187218877082252533, −7.19383089129740938666023607805, −6.64367677864087529475494925552, −6.49574926978295521754769688108, −5.91323595865813664960378916141, −5.58163714663183829670662499015, −4.89680318143745927342182143807, −4.78377900172501744495823396513, −3.57733667604527650710703646354, −3.41648934671731639952703452985, −2.35880976293082522865229911219, −2.06079048662367972003274439679, −1.04556555540668435101019840246, −0.75820356854681945346810287386,
0.75820356854681945346810287386, 1.04556555540668435101019840246, 2.06079048662367972003274439679, 2.35880976293082522865229911219, 3.41648934671731639952703452985, 3.57733667604527650710703646354, 4.78377900172501744495823396513, 4.89680318143745927342182143807, 5.58163714663183829670662499015, 5.91323595865813664960378916141, 6.49574926978295521754769688108, 6.64367677864087529475494925552, 7.19383089129740938666023607805, 7.37109103034187218877082252533, 8.166292991905687590382797175022, 8.362721335182838771891454882484, 9.132533672252278719758903061035, 9.233962082308994120919370484186, 9.785166473869818352386352040370, 9.949991477259471093589151146249
