| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s − 2·5-s + 4·6-s + 4·8-s + 2·9-s − 4·10-s + 2·11-s + 6·12-s + 2·13-s − 4·15-s + 5·16-s + 4·17-s + 4·18-s + 10·19-s − 6·20-s + 4·22-s + 8·23-s + 8·24-s − 2·25-s + 4·26-s + 6·27-s − 8·30-s − 4·31-s + 6·32-s + 4·33-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 + 2/3·9-s − 1.26·10-s + 0.603·11-s + 1.73·12-s + 0.554·13-s − 1.03·15-s + 5/4·16-s + 0.970·17-s + 0.942·18-s + 2.29·19-s − 1.34·20-s + 0.852·22-s + 1.66·23-s + 1.63·24-s − 2/5·25-s + 0.784·26-s + 1.15·27-s − 1.46·30-s − 0.718·31-s + 1.06·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.840023712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.840023712\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 66 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T - 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−10.00829958563360915913838911374, −9.743913546027839964224405485591, −9.142548431920584778533938054409, −8.868440804206357708140504609090, −8.195523686917400799090646390594, −8.107011218752175823817849183982, −7.42804411366250484970207537346, −7.03886884663017761430529089934, −7.03000964499145678562181388468, −6.27285384438401664958786921833, −5.60433448207109467768543833834, −5.28717239674767381762556781599, −4.89768914171678265713289991347, −4.24663132650566531859521090082, −3.69541219660826411839179615129, −3.43146858905407759239798601585, −3.08791371561946288993837862507, −2.65054860399318456182246322213, −1.55719080927706853210503358377, −1.16174622122432376623073397048,
1.16174622122432376623073397048, 1.55719080927706853210503358377, 2.65054860399318456182246322213, 3.08791371561946288993837862507, 3.43146858905407759239798601585, 3.69541219660826411839179615129, 4.24663132650566531859521090082, 4.89768914171678265713289991347, 5.28717239674767381762556781599, 5.60433448207109467768543833834, 6.27285384438401664958786921833, 7.03000964499145678562181388468, 7.03886884663017761430529089934, 7.42804411366250484970207537346, 8.107011218752175823817849183982, 8.195523686917400799090646390594, 8.868440804206357708140504609090, 9.142548431920584778533938054409, 9.743913546027839964224405485591, 10.00829958563360915913838911374
