| L(s) = 1 | + 2-s − 3-s − 4·5-s − 6-s + 7-s − 8-s − 4·10-s − 3·11-s + 7·13-s + 14-s + 4·15-s − 16-s + 7·17-s + 5·19-s − 21-s − 3·22-s − 6·23-s + 24-s + 2·25-s + 7·26-s + 27-s − 5·29-s + 4·30-s − 4·31-s + 3·33-s + 7·34-s − 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1.78·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 1.26·10-s − 0.904·11-s + 1.94·13-s + 0.267·14-s + 1.03·15-s − 1/4·16-s + 1.69·17-s + 1.14·19-s − 0.218·21-s − 0.639·22-s − 1.25·23-s + 0.204·24-s + 2/5·25-s + 1.37·26-s + 0.192·27-s − 0.928·29-s + 0.730·30-s − 0.718·31-s + 0.522·33-s + 1.20·34-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.190641558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.190641558\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 8 T - 7 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 11 T + 32 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−11.06983418179630248415065272205, −10.91869553175966774889983300169, −10.32325243113912091357313005013, −9.678484090225954888735881446096, −9.409708751412211656811195109736, −8.491755179740161768005793488727, −8.269359884234608100526984308962, −7.83248590751686316444717867924, −7.55732539444222454888822464511, −7.11278464674031947396722552039, −6.10907682108428761318302572605, −5.96796075785326318956480746654, −5.31500453266636992787214858619, −5.13424848244794757573149239640, −4.05576567797439646211974910588, −3.95479812152735622769408245243, −3.51591198556472570128822410650, −2.90547099722922512681147123474, −1.65863679382763934301921949707, −0.61532203724314882280337745184,
0.61532203724314882280337745184, 1.65863679382763934301921949707, 2.90547099722922512681147123474, 3.51591198556472570128822410650, 3.95479812152735622769408245243, 4.05576567797439646211974910588, 5.13424848244794757573149239640, 5.31500453266636992787214858619, 5.96796075785326318956480746654, 6.10907682108428761318302572605, 7.11278464674031947396722552039, 7.55732539444222454888822464511, 7.83248590751686316444717867924, 8.269359884234608100526984308962, 8.491755179740161768005793488727, 9.409708751412211656811195109736, 9.678484090225954888735881446096, 10.32325243113912091357313005013, 10.91869553175966774889983300169, 11.06983418179630248415065272205
