| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s + 2·5-s − 4·6-s − 4·8-s + 3·9-s − 4·10-s + 4·11-s + 6·12-s + 6·13-s + 4·15-s + 5·16-s + 4·17-s − 6·18-s + 2·19-s + 6·20-s − 8·22-s − 2·23-s − 8·24-s − 12·26-s + 4·27-s + 8·29-s − 8·30-s − 2·31-s − 6·32-s + 8·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s − 1.63·6-s − 1.41·8-s + 9-s − 1.26·10-s + 1.20·11-s + 1.73·12-s + 1.66·13-s + 1.03·15-s + 5/4·16-s + 0.970·17-s − 1.41·18-s + 0.458·19-s + 1.34·20-s − 1.70·22-s − 0.417·23-s − 1.63·24-s − 2.35·26-s + 0.769·27-s + 1.48·29-s − 1.46·30-s − 0.359·31-s − 1.06·32-s + 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31203396 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.160031326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.160031326\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 40 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 56 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 68 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 74 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 88 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 94 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 144 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 266 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−8.370159091397327517221928707537, −8.144613631700482709790695864603, −7.77881342263580783537092194324, −7.41249565022453588318136105902, −6.83844960426736877329209399406, −6.78293100878370548904768105659, −6.21724544195389816905908708864, −6.12142800388548303899150556797, −5.47871580618740699021403693634, −5.37528618542645004687545087206, −4.49259106654720860817667131500, −4.14499667796194990439423486390, −3.60011672344813857246585783264, −3.44421310018118445318643556061, −2.91086153376119468422861044593, −2.45670756063785416225959055150, −1.92646683010808266699767019063, −1.60511101217048910650506170681, −0.995999568716137529784670826493, −0.870189811176183413772264958285,
0.870189811176183413772264958285, 0.995999568716137529784670826493, 1.60511101217048910650506170681, 1.92646683010808266699767019063, 2.45670756063785416225959055150, 2.91086153376119468422861044593, 3.44421310018118445318643556061, 3.60011672344813857246585783264, 4.14499667796194990439423486390, 4.49259106654720860817667131500, 5.37528618542645004687545087206, 5.47871580618740699021403693634, 6.12142800388548303899150556797, 6.21724544195389816905908708864, 6.78293100878370548904768105659, 6.83844960426736877329209399406, 7.41249565022453588318136105902, 7.77881342263580783537092194324, 8.144613631700482709790695864603, 8.370159091397327517221928707537
