| 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 − 6·11-s − 6·12-s − 4·15-s + 5·16-s + 8·17-s + 6·18-s + 2·19-s + 6·20-s − 12·22-s + 2·23-s − 8·24-s − 2·25-s − 4·27-s − 8·30-s − 4·31-s + 6·32-s + 12·33-s + 16·34-s + 9·36-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 − 1.80·11-s − 1.73·12-s − 1.03·15-s + 5/4·16-s + 1.94·17-s + 1.41·18-s + 0.458·19-s + 1.34·20-s − 2.55·22-s + 0.417·23-s − 1.63·24-s − 2/5·25-s − 0.769·27-s − 1.46·30-s − 0.718·31-s + 1.06·32-s + 2.08·33-s + 2.74·34-s + 3/2·36-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
| \(L(1)\) |
\(\approx\) |
\(5.871843106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.871843106\) |
| \(L(\frac{3}{2})\) |
|
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 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_4$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 18 T + 150 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 126 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 22 T + 282 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_4$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.85176760319388738683138933320, −7.74411312925280007893831169427, −7.28223723235191053100342360186, −7.11598081722120877698371690495, −6.49283624411354344663123252064, −6.12880574620820230116965418109, −5.93640054438167707485972734348, −5.65035145052256515520646537374, −5.27470456384454945037502439596, −5.16710467432510798123569150681, −4.64647748014847667139688059679, −4.48809975862707633011525351641, −3.75225692261898516959088259902, −3.46807072122014520108131867037, −2.91926655307911333168944879329, −2.73166704462028131537592896456, −2.09816380183877348173326297853, −1.68827362115027907319905257884, −1.11892676481292312876691637756, −0.52016496873577664065999582433,
0.52016496873577664065999582433, 1.11892676481292312876691637756, 1.68827362115027907319905257884, 2.09816380183877348173326297853, 2.73166704462028131537592896456, 2.91926655307911333168944879329, 3.46807072122014520108131867037, 3.75225692261898516959088259902, 4.48809975862707633011525351641, 4.64647748014847667139688059679, 5.16710467432510798123569150681, 5.27470456384454945037502439596, 5.65035145052256515520646537374, 5.93640054438167707485972734348, 6.12880574620820230116965418109, 6.49283624411354344663123252064, 7.11598081722120877698371690495, 7.28223723235191053100342360186, 7.74411312925280007893831169427, 7.85176760319388738683138933320
