| L(s) = 1 | − 3·2-s − 3-s + 4·4-s + 2·5-s + 3·6-s + 2·7-s − 3·8-s − 4·9-s − 6·10-s − 4·12-s + 2·13-s − 6·14-s − 2·15-s + 3·16-s − 5·17-s + 12·18-s − 8·19-s + 8·20-s − 2·21-s − 23-s + 3·24-s − 7·25-s − 6·26-s + 6·27-s + 8·28-s − 7·29-s + 6·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 2·4-s + 0.894·5-s + 1.22·6-s + 0.755·7-s − 1.06·8-s − 4/3·9-s − 1.89·10-s − 1.15·12-s + 0.554·13-s − 1.60·14-s − 0.516·15-s + 3/4·16-s − 1.21·17-s + 2.82·18-s − 1.83·19-s + 1.78·20-s − 0.436·21-s − 0.208·23-s + 0.612·24-s − 7/5·25-s − 1.17·26-s + 1.15·27-s + 1.51·28-s − 1.29·29-s + 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 717409 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 717409 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T + 15 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 7 T + 69 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 61 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 43 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T - 9 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 123 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 117 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 163 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 73 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 21 T + 221 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 24 T + 285 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 15 T + 213 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 137 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 179 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−9.865078960704782089208818551945, −9.482838818025454908029145363388, −8.984160913290049994533083852919, −8.876017411837362029824939682435, −8.366164039787206267798900311225, −8.191099614969721863271154062532, −7.61495633123750741670707522513, −7.22059825385241821925502660797, −6.39173109338585073287907499930, −6.05112249494461483758841946808, −5.97163257254285904479895135582, −5.30398997807561911981799181329, −4.67318066006327461208361749261, −4.11489933847878249489477622553, −3.33236426741763899825414396496, −2.48188325004143288463047994564, −1.85083387354783508676328934534, −1.55579214088232074399455203926, 0, 0,
1.55579214088232074399455203926, 1.85083387354783508676328934534, 2.48188325004143288463047994564, 3.33236426741763899825414396496, 4.11489933847878249489477622553, 4.67318066006327461208361749261, 5.30398997807561911981799181329, 5.97163257254285904479895135582, 6.05112249494461483758841946808, 6.39173109338585073287907499930, 7.22059825385241821925502660797, 7.61495633123750741670707522513, 8.191099614969721863271154062532, 8.366164039787206267798900311225, 8.876017411837362029824939682435, 8.984160913290049994533083852919, 9.482838818025454908029145363388, 9.865078960704782089208818551945
