| 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 + 2·13-s − 4·15-s + 5·16-s + 2·17-s − 6·18-s − 4·19-s − 6·20-s − 8·22-s + 2·23-s − 8·24-s − 5·25-s − 4·26-s + 4·27-s − 8·29-s + 8·30-s − 4·31-s − 6·32-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 + 0.554·13-s − 1.03·15-s + 5/4·16-s + 0.485·17-s − 1.41·18-s − 0.917·19-s − 1.34·20-s − 1.70·22-s + 0.417·23-s − 1.63·24-s − 25-s − 0.784·26-s + 0.769·27-s − 1.48·29-s + 1.46·30-s − 0.718·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45724644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T + 65 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 61 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 14 T + 183 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 139 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 258 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 8 T + 186 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 178 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.67759728705719140687677062843, −7.62901092275305047918403564746, −7.27987235799357676215575473664, −7.04300348232102476869604794519, −6.50517748115669648877602271276, −6.38151723442507486773653407395, −5.70596934263359482450978427991, −5.56087323196958118367532855392, −4.76996647416616588256676105983, −4.52121356758867834783235289540, −3.78330475745964104650034654622, −3.65716774358732905653342793946, −3.43023394982347744072956813846, −3.08448926341844200250839977250, −2.10625145057678674976717585327, −2.09878547812877382662816248002, −1.43680428425558666023060059562, −1.27271154167007686225121559356, 0, 0,
1.27271154167007686225121559356, 1.43680428425558666023060059562, 2.09878547812877382662816248002, 2.10625145057678674976717585327, 3.08448926341844200250839977250, 3.43023394982347744072956813846, 3.65716774358732905653342793946, 3.78330475745964104650034654622, 4.52121356758867834783235289540, 4.76996647416616588256676105983, 5.56087323196958118367532855392, 5.70596934263359482450978427991, 6.38151723442507486773653407395, 6.50517748115669648877602271276, 7.04300348232102476869604794519, 7.27987235799357676215575473664, 7.62901092275305047918403564746, 7.67759728705719140687677062843
