| L(s) = 1 | − 8·2-s + 48·4-s − 50·5-s + 98·7-s − 256·8-s + 400·10-s − 415·11-s + 429·13-s − 784·14-s + 1.28e3·16-s − 1.31e3·17-s + 1.91e3·19-s − 2.40e3·20-s + 3.32e3·22-s + 1.33e3·23-s + 1.87e3·25-s − 3.43e3·26-s + 4.70e3·28-s + 1.13e3·29-s − 5.47e3·31-s − 6.14e3·32-s + 1.05e4·34-s − 4.90e3·35-s − 9.15e3·37-s − 1.53e4·38-s + 1.28e4·40-s + 7.82e3·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 0.755·7-s − 1.41·8-s + 1.26·10-s − 1.03·11-s + 0.704·13-s − 1.06·14-s + 5/4·16-s − 1.10·17-s + 1.21·19-s − 1.34·20-s + 1.46·22-s + 0.525·23-s + 3/5·25-s − 0.995·26-s + 1.13·28-s + 0.249·29-s − 1.02·31-s − 1.06·32-s + 1.56·34-s − 0.676·35-s − 1.09·37-s − 1.72·38-s + 1.26·40-s + 0.726·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 + 415 T + 317458 T^{2} + 415 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 33 p T + 754444 T^{2} - 33 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1319 T + 2082148 T^{2} + 1319 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1918 T + 2203758 T^{2} - 1918 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 58 p T + 10606846 T^{2} - 58 p^{6} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 39 p T + 38900908 T^{2} - 39 p^{6} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5472 T + 8095294 T^{2} + 5472 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 9156 T + 152420398 T^{2} + 9156 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7822 T + 243078274 T^{2} - 7822 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2078 T - 70429762 T^{2} - 2078 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 181 p T - 114195314 T^{2} - 181 p^{6} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 34242 T + 846139498 T^{2} - 34242 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6952 T + 1381163878 T^{2} + 6952 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 45138 T + 2092145242 T^{2} + 45138 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 54556 T + 3349390598 T^{2} + 54556 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 62272 T + 3502450894 T^{2} - 62272 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 40896 T + 3740866126 T^{2} - 40896 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 62069 T + 5500962758 T^{2} - 62069 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 189600 T + 16864958710 T^{2} + 189600 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 64190 T + 10032531898 T^{2} + 64190 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 155547 T + 22978754524 T^{2} - 155547 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.326712880359461507121106808545, −9.180688583227502416131628132458, −8.632688983827928196838340612684, −8.438683498760250910717458182594, −7.81390934947602375222277276298, −7.58071303606908901384195881695, −7.10204264645172474694690669705, −6.89080306650469333196746931726, −5.91646242295858855613505626012, −5.76576792618553472644060161405, −4.85560805878938820665457910768, −4.73099295799297464266473822669, −3.62140056002982387592423351857, −3.53137253624999977630538087382, −2.44203452079553159055582964029, −2.38063254103515723393381437780, −1.19372965541699787241538317876, −1.16317544232529964266850267615, 0, 0,
1.16317544232529964266850267615, 1.19372965541699787241538317876, 2.38063254103515723393381437780, 2.44203452079553159055582964029, 3.53137253624999977630538087382, 3.62140056002982387592423351857, 4.73099295799297464266473822669, 4.85560805878938820665457910768, 5.76576792618553472644060161405, 5.91646242295858855613505626012, 6.89080306650469333196746931726, 7.10204264645172474694690669705, 7.58071303606908901384195881695, 7.81390934947602375222277276298, 8.438683498760250910717458182594, 8.632688983827928196838340612684, 9.180688583227502416131628132458, 9.326712880359461507121106808545
