| L(s) = 1 | − 6·7-s − 2·11-s − 13-s + 3·17-s − 2·19-s + 5·23-s − 4·29-s − 31-s − 5·37-s − 11·41-s + 9·43-s + 9·47-s + 13·49-s − 4·53-s − 9·59-s − 5·61-s − 19·67-s − 71-s + 28·73-s + 12·77-s + 79-s + 2·83-s − 12·89-s + 6·91-s + 4·97-s − 23·101-s + 32·103-s + ⋯ |
| L(s) = 1 | − 2.26·7-s − 0.603·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s + 1.04·23-s − 0.742·29-s − 0.179·31-s − 0.821·37-s − 1.71·41-s + 1.37·43-s + 1.31·47-s + 13/7·49-s − 0.549·53-s − 1.17·59-s − 0.640·61-s − 2.32·67-s − 0.118·71-s + 3.27·73-s + 1.36·77-s + 0.112·79-s + 0.219·83-s − 1.27·89-s + 0.628·91-s + 0.406·97-s − 2.28·101-s + 3.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.436909685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.436909685\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + T + 26 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 44 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 11 T + 76 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 9 T + 70 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 9 T + 102 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 92 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 19 T + 188 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T + 106 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - T + 122 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 53 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.65972448676120221246927858819, −7.55572206294096662981293449258, −7.05140345369512539734715255718, −6.83414741057502698206990049044, −6.38690322296747967219850036184, −6.32026944799558943342136203881, −5.76878050227825717492070701147, −5.54636427288980180403192355510, −5.09235423474660880558350294040, −4.89088749307486927574310891422, −4.11656516459774508360728063635, −4.08229140047241746699810785944, −3.33339653849628678272920455198, −3.32106111586384358443960742829, −2.80124274311027473518608581677, −2.71231448767608301340975118770, −1.80846464064135687426206379459, −1.67479594780627922710665985953, −0.59431718844295663306075109652, −0.43299779588249009354639598568,
0.43299779588249009354639598568, 0.59431718844295663306075109652, 1.67479594780627922710665985953, 1.80846464064135687426206379459, 2.71231448767608301340975118770, 2.80124274311027473518608581677, 3.32106111586384358443960742829, 3.33339653849628678272920455198, 4.08229140047241746699810785944, 4.11656516459774508360728063635, 4.89088749307486927574310891422, 5.09235423474660880558350294040, 5.54636427288980180403192355510, 5.76878050227825717492070701147, 6.32026944799558943342136203881, 6.38690322296747967219850036184, 6.83414741057502698206990049044, 7.05140345369512539734715255718, 7.55572206294096662981293449258, 7.65972448676120221246927858819
