| L(s) = 1 | − 2·2-s + 4·3-s + 4-s + 2·5-s − 8·6-s + 8·9-s − 4·10-s − 8·11-s + 4·12-s + 8·15-s + 16-s + 2·17-s − 16·18-s + 2·20-s + 16·22-s − 4·23-s + 3·25-s + 12·27-s − 4·29-s − 16·30-s + 2·32-s − 32·33-s − 4·34-s + 8·36-s − 4·37-s − 4·41-s + 4·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 1/2·4-s + 0.894·5-s − 3.26·6-s + 8/3·9-s − 1.26·10-s − 2.41·11-s + 1.15·12-s + 2.06·15-s + 1/4·16-s + 0.485·17-s − 3.77·18-s + 0.447·20-s + 3.41·22-s − 0.834·23-s + 3/5·25-s + 2.30·27-s − 0.742·29-s − 2.92·30-s + 0.353·32-s − 5.57·33-s − 0.685·34-s + 4/3·36-s − 0.657·37-s − 0.624·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17347225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17347225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.019513164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.019513164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 8 T + 36 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 24 T + 254 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 94 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 124 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 172 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 210 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.471334596153760723031639412360, −8.416716391038997407877224425800, −8.024273654407715665244808131934, −7.79560010234890287479082111417, −7.38102209599805257057261034853, −7.21005593322946190498962717405, −6.56356562205533769680672604943, −6.10007406137918267592228596715, −5.53905242158204754746284993800, −5.34173370997925694572930967407, −4.94219050165446303731289739669, −4.30211212707481108770029317800, −3.79465980502458632094007901229, −3.34449411688201667767396946334, −3.00220283623335111817960833023, −2.46026625179740911005493246561, −2.24535848289359650101831310031, −2.00216227959192056755712169015, −1.12052162663156479128229003231, −0.45910833576866128376978929856,
0.45910833576866128376978929856, 1.12052162663156479128229003231, 2.00216227959192056755712169015, 2.24535848289359650101831310031, 2.46026625179740911005493246561, 3.00220283623335111817960833023, 3.34449411688201667767396946334, 3.79465980502458632094007901229, 4.30211212707481108770029317800, 4.94219050165446303731289739669, 5.34173370997925694572930967407, 5.53905242158204754746284993800, 6.10007406137918267592228596715, 6.56356562205533769680672604943, 7.21005593322946190498962717405, 7.38102209599805257057261034853, 7.79560010234890287479082111417, 8.024273654407715665244808131934, 8.416716391038997407877224425800, 8.471334596153760723031639412360
