| L(s) = 1 | − 8·2-s − 5·3-s + 48·4-s + 40·6-s − 98·7-s − 256·8-s − 185·9-s + 415·11-s − 240·12-s − 429·13-s + 784·14-s + 1.28e3·16-s − 1.31e3·17-s + 1.48e3·18-s + 1.91e3·19-s + 490·21-s − 3.32e3·22-s + 1.33e3·23-s + 1.28e3·24-s + 3.43e3·26-s + 760·27-s − 4.70e3·28-s − 1.13e3·29-s − 5.47e3·31-s − 6.14e3·32-s − 2.07e3·33-s + 1.05e4·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.320·3-s + 3/2·4-s + 0.453·6-s − 0.755·7-s − 1.41·8-s − 0.761·9-s + 1.03·11-s − 0.481·12-s − 0.704·13-s + 1.06·14-s + 5/4·16-s − 1.10·17-s + 1.07·18-s + 1.21·19-s + 0.242·21-s − 1.46·22-s + 0.525·23-s + 0.453·24-s + 0.995·26-s + 0.200·27-s − 1.13·28-s − 0.249·29-s − 1.02·31-s − 1.06·32-s − 0.331·33-s + 1.56·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 5 T + 70 p T^{2} + 5 p^{5} T^{3} + p^{10} T^{4} \) |
| 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
−10.38868304814197244864684583032, −9.743125265180136967832348672775, −9.411075551449307212161340356311, −9.263055503283541779605107291392, −8.565342975898545369161051037890, −8.355236588551091043266516771624, −7.41519047390506433311844203793, −7.23563766348301664090699992769, −6.72512503625368285468796041641, −6.28773082868192240715837114685, −5.58787527741936466887580722342, −5.33125099248837841779382226465, −4.26467838631811684402604213290, −3.73579329572425449819983081828, −2.77685717157013274585504820013, −2.64812041274584959464628672297, −1.57465376191117731310130598059, −1.04486655230655723200970365918, 0, 0,
1.04486655230655723200970365918, 1.57465376191117731310130598059, 2.64812041274584959464628672297, 2.77685717157013274585504820013, 3.73579329572425449819983081828, 4.26467838631811684402604213290, 5.33125099248837841779382226465, 5.58787527741936466887580722342, 6.28773082868192240715837114685, 6.72512503625368285468796041641, 7.23563766348301664090699992769, 7.41519047390506433311844203793, 8.355236588551091043266516771624, 8.565342975898545369161051037890, 9.263055503283541779605107291392, 9.411075551449307212161340356311, 9.743125265180136967832348672775, 10.38868304814197244864684583032
