| L(s) = 1 | − 6·3-s + 18·5-s − 10·7-s + 27·9-s − 4·11-s − 26·13-s − 108·15-s − 52·17-s + 142·19-s + 60·21-s − 280·23-s + 10·25-s − 108·27-s + 424·29-s − 178·31-s + 24·33-s − 180·35-s − 136·37-s + 156·39-s − 226·41-s + 72·43-s + 486·45-s + 176·47-s − 186·49-s + 312·51-s + 788·53-s − 72·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.60·5-s − 0.539·7-s + 9-s − 0.109·11-s − 0.554·13-s − 1.85·15-s − 0.741·17-s + 1.71·19-s + 0.623·21-s − 2.53·23-s + 2/25·25-s − 0.769·27-s + 2.71·29-s − 1.03·31-s + 0.126·33-s − 0.869·35-s − 0.604·37-s + 0.640·39-s − 0.860·41-s + 0.255·43-s + 1.60·45-s + 0.546·47-s − 0.542·49-s + 0.856·51-s + 2.04·53-s − 0.176·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7916360677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7916360677\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 18 T + 314 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 10 T + 286 T^{2} + 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T - 666 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 52 T + 8054 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 142 T + 16702 T^{2} - 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 280 T + 41486 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 424 T + 93110 T^{2} - 424 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 178 T + 48990 T^{2} + 178 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 136 T + 56358 T^{2} + 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 226 T + 144474 T^{2} + 226 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 72 T + 28662 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 176 T - 29410 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 788 T + 451902 T^{2} - 788 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 728 T + 542166 T^{2} + 728 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 736 T + 564838 T^{2} - 736 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1054 T + 684622 T^{2} + 1054 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 660 T + 721294 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 24 T - 59650 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 40 T - 308514 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1344 T + 1516550 T^{2} + 1344 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 314 T + 619250 T^{2} - 314 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 264 T + 1841070 T^{2} + 264 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.663040339537159996800988353922, −8.637322623139489185634022187790, −7.77389835717044415361569491672, −7.65169715757925005453971726819, −6.94948595431688625402849838888, −6.88645624154527224327976326832, −6.22274566693002830088751709379, −6.04468920200788309190695031104, −5.70595912021884655323917946443, −5.46684814163672059634228752514, −4.87653997649164264083503781619, −4.61273054487641761704473496898, −3.99933366716651493691274307760, −3.60574065369681497554899255004, −2.81771518075726149318426536456, −2.53772379526104580160829602345, −1.82099928139287013288395700752, −1.62939953893538085647119128250, −0.893979176437509802330674656599, −0.20963803074606327814336657667,
0.20963803074606327814336657667, 0.893979176437509802330674656599, 1.62939953893538085647119128250, 1.82099928139287013288395700752, 2.53772379526104580160829602345, 2.81771518075726149318426536456, 3.60574065369681497554899255004, 3.99933366716651493691274307760, 4.61273054487641761704473496898, 4.87653997649164264083503781619, 5.46684814163672059634228752514, 5.70595912021884655323917946443, 6.04468920200788309190695031104, 6.22274566693002830088751709379, 6.88645624154527224327976326832, 6.94948595431688625402849838888, 7.65169715757925005453971726819, 7.77389835717044415361569491672, 8.637322623139489185634022187790, 8.663040339537159996800988353922
