| L(s) = 1 | + 10·2-s + 50·4-s − 80·7-s + 160·8-s − 200·11-s − 410·13-s − 800·14-s + 444·16-s − 470·17-s − 2.00e3·22-s − 680·23-s − 4.10e3·26-s − 4.00e3·28-s + 856·31-s + 1.88e3·32-s − 4.70e3·34-s + 1.51e3·37-s + 1.90e3·41-s + 2.44e3·43-s − 1.00e4·44-s − 6.80e3·46-s + 640·47-s + 3.20e3·49-s − 2.05e4·52-s − 1.01e3·53-s − 1.28e4·56-s − 7.61e3·61-s + ⋯ |
| L(s) = 1 | + 5/2·2-s + 25/8·4-s − 1.63·7-s + 5/2·8-s − 1.65·11-s − 2.42·13-s − 4.08·14-s + 1.73·16-s − 1.62·17-s − 4.13·22-s − 1.28·23-s − 6.06·26-s − 5.10·28-s + 0.890·31-s + 1.83·32-s − 4.06·34-s + 1.10·37-s + 1.13·41-s + 1.31·43-s − 5.16·44-s − 3.21·46-s + 0.289·47-s + 1.33·49-s − 7.58·52-s − 0.359·53-s − 4.08·56-s − 2.04·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.333682973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.333682973\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 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 | $C_2^2$ | \( 1 + 3230 T + 5216450 T^{2} + 3230 p^{4} T^{3} + p^{8} T^{4} \) |
| 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
−12.55421471218701869351066010068, −11.69213355904975717987345693736, −11.06389114268157560372897216401, −10.26974151151785631110932435666, −10.23430204421217441767462945683, −9.487072320517781640050936444603, −9.119588414941252156007962277407, −7.982799949614261600375029719650, −7.60390544468379299004763625416, −7.10114394777268596358748759672, −6.26406391894852383770534352395, −6.13959574203618738266371137518, −5.43413550648884262308046698042, −4.89533719470764430401358474290, −4.35928677832489144257749067065, −4.08933322307680087666755610289, −2.92995721537934280711402688583, −2.72800090046431820289679764458, −2.31446904081383294403819208594, −0.23231090340722660852265594036,
0.23231090340722660852265594036, 2.31446904081383294403819208594, 2.72800090046431820289679764458, 2.92995721537934280711402688583, 4.08933322307680087666755610289, 4.35928677832489144257749067065, 4.89533719470764430401358474290, 5.43413550648884262308046698042, 6.13959574203618738266371137518, 6.26406391894852383770534352395, 7.10114394777268596358748759672, 7.60390544468379299004763625416, 7.982799949614261600375029719650, 9.119588414941252156007962277407, 9.487072320517781640050936444603, 10.23430204421217441767462945683, 10.26974151151785631110932435666, 11.06389114268157560372897216401, 11.69213355904975717987345693736, 12.55421471218701869351066010068
