| L(s) = 1 | + 4-s + 4·5-s + 2·7-s + 2·11-s + 2·13-s − 3·16-s + 2·17-s + 4·19-s + 4·20-s + 4·23-s + 2·25-s + 2·28-s − 8·29-s − 10·31-s + 8·35-s − 8·37-s + 18·41-s + 16·43-s + 2·44-s − 10·47-s + 3·49-s + 2·52-s − 8·53-s + 8·55-s − 2·59-s − 10·61-s − 7·64-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.78·5-s + 0.755·7-s + 0.603·11-s + 0.554·13-s − 3/4·16-s + 0.485·17-s + 0.917·19-s + 0.894·20-s + 0.834·23-s + 2/5·25-s + 0.377·28-s − 1.48·29-s − 1.79·31-s + 1.35·35-s − 1.31·37-s + 2.81·41-s + 2.43·43-s + 0.301·44-s − 1.45·47-s + 3/7·49-s + 0.277·52-s − 1.09·53-s + 1.07·55-s − 0.260·59-s − 1.28·61-s − 7/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.749348001\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.749348001\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 2 T + 114 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 150 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.88226261751898714785691828972, −10.22111387886137862947032289887, −9.618661278723839710935866831226, −9.445859762604962707735678605681, −9.030420752058569698035394175085, −8.875903203689607430711580846881, −7.906753113591387314188233319398, −7.48512361075406458516301979045, −7.35617237106010774555576331066, −6.55903344230073039738978596285, −6.15276351439606441689623798640, −5.78469681677328770050691654018, −5.36978908974195260336942213258, −4.99203528635377175857833357849, −4.12946067720582032634556591522, −3.67146444257035982143223456680, −2.90741800162899026863915305070, −2.09099019589705569255621025853, −1.86016951804562747704185740948, −1.13464446269756347973552304810,
1.13464446269756347973552304810, 1.86016951804562747704185740948, 2.09099019589705569255621025853, 2.90741800162899026863915305070, 3.67146444257035982143223456680, 4.12946067720582032634556591522, 4.99203528635377175857833357849, 5.36978908974195260336942213258, 5.78469681677328770050691654018, 6.15276351439606441689623798640, 6.55903344230073039738978596285, 7.35617237106010774555576331066, 7.48512361075406458516301979045, 7.906753113591387314188233319398, 8.875903203689607430711580846881, 9.030420752058569698035394175085, 9.445859762604962707735678605681, 9.618661278723839710935866831226, 10.22111387886137862947032289887, 10.88226261751898714785691828972
