| L(s) = 1 | + 10·2-s + 50·4-s − 50·5-s + 80·7-s + 160·8-s − 500·10-s + 200·11-s + 410·13-s + 800·14-s + 444·16-s − 470·17-s − 2.50e3·20-s + 2.00e3·22-s − 680·23-s + 1.87e3·25-s + 4.10e3·26-s + 4.00e3·28-s + 856·31-s + 1.88e3·32-s − 4.70e3·34-s − 4.00e3·35-s − 1.51e3·37-s − 8.00e3·40-s − 1.90e3·41-s − 2.44e3·43-s + 1.00e4·44-s − 6.80e3·46-s + ⋯ |
| L(s) = 1 | + 5/2·2-s + 25/8·4-s − 2·5-s + 1.63·7-s + 5/2·8-s − 5·10-s + 1.65·11-s + 2.42·13-s + 4.08·14-s + 1.73·16-s − 1.62·17-s − 6.25·20-s + 4.13·22-s − 1.28·23-s + 3·25-s + 6.06·26-s + 5.10·28-s + 0.890·31-s + 1.83·32-s − 4.06·34-s − 3.26·35-s − 1.10·37-s − 5·40-s − 1.13·41-s − 1.31·43-s + 5.16·44-s − 3.21·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(6.653285197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.653285197\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 5 p T + 25 p T^{2} - 5 p^{5} T^{3} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 80 T + 3200 T^{2} - 80 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 100 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 410 T + 84050 T^{2} - 410 p^{4} T^{3} + p^{8} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 470 T + 110450 T^{2} + 470 p^{4} T^{3} + p^{8} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 255458 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 680 T + 231200 T^{2} + 680 p^{4} T^{3} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1212062 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 428 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 1510 T + 1140050 T^{2} + 1510 p^{4} T^{3} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 950 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 2440 T + 2976800 T^{2} + 2440 p^{4} T^{3} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 640 T + 204800 T^{2} - 640 p^{4} T^{3} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1010 T + 510050 T^{2} + 1010 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 15455278 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3808 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 680 T + 231200 T^{2} - 680 p^{4} T^{3} + p^{8} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3400 T + p^{4} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 830 T + 344450 T^{2} - 830 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 32580338 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1360 T + 924800 T^{2} - 1360 p^{4} T^{3} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 120421982 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 3230 T + 5216450 T^{2} - 3230 p^{4} T^{3} + p^{8} T^{4} \) |
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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
−15.50133925240445655956690304286, −14.55184907174678637870667255942, −14.29068522875328337188023796455, −13.53701589646932624912826224300, −13.43245594929028086832868376588, −12.17659912175821356236754410929, −12.13149936916721339299296009676, −11.34315914346371022995727218886, −11.33027201352804955914988420686, −10.60438310806087367317294542635, −8.617773990576936694291157365692, −8.612532198335473653513341754148, −7.75145767661683963319486832515, −6.60452755635537267947732298830, −6.26719112180111702936452251074, −4.93693590769966922850743753299, −4.48070873773314034949703832178, −3.81635200927165566925464618260, −3.56099129289252301819949620407, −1.46122874547668141519099399848,
1.46122874547668141519099399848, 3.56099129289252301819949620407, 3.81635200927165566925464618260, 4.48070873773314034949703832178, 4.93693590769966922850743753299, 6.26719112180111702936452251074, 6.60452755635537267947732298830, 7.75145767661683963319486832515, 8.612532198335473653513341754148, 8.617773990576936694291157365692, 10.60438310806087367317294542635, 11.33027201352804955914988420686, 11.34315914346371022995727218886, 12.13149936916721339299296009676, 12.17659912175821356236754410929, 13.43245594929028086832868376588, 13.53701589646932624912826224300, 14.29068522875328337188023796455, 14.55184907174678637870667255942, 15.50133925240445655956690304286
