| L(s) = 1 | − 14·3-s − 50·5-s + 164·7-s − 272·9-s + 162·11-s + 338·13-s + 700·15-s + 2.00e3·17-s − 858·19-s − 2.29e3·21-s − 1.53e3·23-s + 1.87e3·25-s + 6.95e3·27-s + 740·29-s − 2.05e3·31-s − 2.26e3·33-s − 8.20e3·35-s + 7.52e3·37-s − 4.73e3·39-s − 1.26e4·41-s − 2.50e4·43-s + 1.36e4·45-s − 2.69e4·47-s − 1.23e4·49-s − 2.81e4·51-s − 1.98e4·53-s − 8.10e3·55-s + ⋯ |
| L(s) = 1 | − 0.898·3-s − 0.894·5-s + 1.26·7-s − 1.11·9-s + 0.403·11-s + 0.554·13-s + 0.803·15-s + 1.68·17-s − 0.545·19-s − 1.13·21-s − 0.606·23-s + 3/5·25-s + 1.83·27-s + 0.163·29-s − 0.383·31-s − 0.362·33-s − 1.13·35-s + 0.903·37-s − 0.498·39-s − 1.17·41-s − 2.06·43-s + 1.00·45-s − 1.77·47-s − 0.736·49-s − 1.51·51-s − 0.970·53-s − 0.361·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67600 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 14 T + 52 p^{2} T^{2} + 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 164 T + 39266 T^{2} - 164 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 162 T + 226756 T^{2} - 162 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2008 T + 3255718 T^{2} - 2008 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 858 T + 4652164 T^{2} + 858 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 1538 T + 5303380 T^{2} + 1538 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 740 T + 19997650 T^{2} - 740 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2054 T - 14293596 T^{2} + 2054 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7520 T + 143150714 T^{2} - 7520 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12664 T + 209801878 T^{2} + 12664 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 25078 T + 350043860 T^{2} + 25078 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 26900 T + 587234962 T^{2} + 26900 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 19848 T + 885553774 T^{2} + 19848 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 21662 T + 1535951332 T^{2} + 21662 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 30108 T + 1913692690 T^{2} + 30108 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 30936 T + 2912237290 T^{2} + 30936 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 23958 T + 2429358460 T^{2} - 23958 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 101328 T + 6699295714 T^{2} + 101328 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 22716 T + 3117241270 T^{2} + 22716 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 20316 T + 7501480858 T^{2} + 20316 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 41116 T + 2102880262 T^{2} + 41116 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 158076 T + 23113704646 T^{2} + 158076 p^{5} T^{3} + p^{10} T^{4} \) |
| 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
−10.90815919640014710750977510393, −10.83990445098406229988802646814, −9.965005976572969729031237353807, −9.603159484616729216167599041623, −8.613382382350082155655909386006, −8.472413083096516399727608161981, −7.87993883232018307048022654278, −7.79256654791818174079597910658, −6.60906165971045000790503756964, −6.51525788172676977554537037022, −5.58286795965957143679341336888, −5.41963302891545621952467093763, −4.67831238675472392119116139761, −4.25532120048384809056863399805, −3.17142304473618758255654242242, −3.13138800565247610170293636885, −1.59595684159444574416356070895, −1.34052019334809229436127989774, 0, 0,
1.34052019334809229436127989774, 1.59595684159444574416356070895, 3.13138800565247610170293636885, 3.17142304473618758255654242242, 4.25532120048384809056863399805, 4.67831238675472392119116139761, 5.41963302891545621952467093763, 5.58286795965957143679341336888, 6.51525788172676977554537037022, 6.60906165971045000790503756964, 7.79256654791818174079597910658, 7.87993883232018307048022654278, 8.472413083096516399727608161981, 8.613382382350082155655909386006, 9.603159484616729216167599041623, 9.965005976572969729031237353807, 10.83990445098406229988802646814, 10.90815919640014710750977510393
