| L(s) = 1 | − 10·5-s − 8·7-s + 64·11-s + 12·13-s − 4·17-s − 208·19-s + 120·23-s + 75·25-s + 292·29-s − 176·31-s + 80·35-s − 356·37-s − 100·41-s + 376·43-s − 280·47-s − 38·49-s − 316·53-s − 640·55-s + 720·59-s − 1.26e3·61-s − 120·65-s + 744·67-s − 48·71-s − 940·73-s − 512·77-s − 32·79-s − 1.59e3·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.431·7-s + 1.75·11-s + 0.256·13-s − 0.0570·17-s − 2.51·19-s + 1.08·23-s + 3/5·25-s + 1.86·29-s − 1.01·31-s + 0.386·35-s − 1.58·37-s − 0.380·41-s + 1.33·43-s − 0.868·47-s − 0.110·49-s − 0.818·53-s − 1.56·55-s + 1.58·59-s − 2.66·61-s − 0.228·65-s + 1.35·67-s − 0.0802·71-s − 1.50·73-s − 0.757·77-s − 0.0455·79-s − 2.10·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.404626744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.404626744\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 64 T + 2822 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 12 T + 2894 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T - 3994 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 208 T + 22998 T^{2} + 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 120 T + 25030 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 292 T + 63950 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 176 T + 66462 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 356 T + 126846 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 100 T + 101942 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 376 T + 161502 T^{2} - 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 280 T + 223190 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 316 T + 308894 T^{2} + 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 720 T + 530758 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1268 T + 831342 T^{2} + 1268 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 744 T + 686894 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 48 T - 424178 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 940 T + 653334 T^{2} + 940 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 32 T + 276318 T^{2} + 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1592 T + 1724174 T^{2} + 1592 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 780 T + 1506742 T^{2} - 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1220 T + 2159046 T^{2} - 1220 p^{3} T^{3} + p^{6} T^{4} \) |
| 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
−8.973801619874975540379668426171, −8.935767511904099718001182259529, −8.711478579753449418536952165435, −8.369270924290798597498347477850, −7.69974727351707514899423286269, −7.29102743881890859235277841509, −6.76799234624203607251051285552, −6.68204654785705396439806314942, −6.10331448735096360129436861705, −5.95382240961210189845530085716, −4.84696008836074368626689389313, −4.83175487950912518280131147525, −4.05064931601639028932095118005, −4.03307804656731597282123137115, −3.24952369985520529897544702800, −3.05372288131847205122141959279, −2.12912958084705137500288520384, −1.65180196271476136767296218214, −0.970065573736024977370780967225, −0.31248267891374571895479596989,
0.31248267891374571895479596989, 0.970065573736024977370780967225, 1.65180196271476136767296218214, 2.12912958084705137500288520384, 3.05372288131847205122141959279, 3.24952369985520529897544702800, 4.03307804656731597282123137115, 4.05064931601639028932095118005, 4.83175487950912518280131147525, 4.84696008836074368626689389313, 5.95382240961210189845530085716, 6.10331448735096360129436861705, 6.68204654785705396439806314942, 6.76799234624203607251051285552, 7.29102743881890859235277841509, 7.69974727351707514899423286269, 8.369270924290798597498347477850, 8.711478579753449418536952165435, 8.935767511904099718001182259529, 8.973801619874975540379668426171
