| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 4·8-s − 4·10-s − 2·11-s + 2·13-s + 5·16-s − 4·17-s + 10·19-s + 6·20-s + 4·22-s − 8·23-s − 2·25-s − 4·26-s − 4·31-s − 6·32-s + 8·34-s − 4·37-s − 20·38-s − 8·40-s + 4·41-s − 12·43-s − 6·44-s + 16·46-s − 4·47-s + 4·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 1.41·8-s − 1.26·10-s − 0.603·11-s + 0.554·13-s + 5/4·16-s − 0.970·17-s + 2.29·19-s + 1.34·20-s + 0.852·22-s − 1.66·23-s − 2/5·25-s − 0.784·26-s − 0.718·31-s − 1.06·32-s + 1.37·34-s − 0.657·37-s − 3.24·38-s − 1.26·40-s + 0.624·41-s − 1.82·43-s − 0.904·44-s + 2.35·46-s − 0.583·47-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94128804 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 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
−7.50981064104359983985970421638, −7.27106900226214349429361205246, −6.87510090786233085536772419350, −6.67163897016025209483958688862, −6.11576293597130253520701997911, −5.92388809734210315233999904073, −5.52652317050472845038906012765, −5.39039573573711553871805384457, −4.77443602414202535048391254339, −4.50968305964953760476989565570, −3.69928742332344896828782178825, −3.55777187437865458875377950042, −3.16579956403131912652733825513, −2.58674712908158369260407786845, −2.16452390180340136583866657341, −1.95391792339217301259235669777, −1.32155024583528464959673079686, −1.13443543830627442846102815885, 0, 0,
1.13443543830627442846102815885, 1.32155024583528464959673079686, 1.95391792339217301259235669777, 2.16452390180340136583866657341, 2.58674712908158369260407786845, 3.16579956403131912652733825513, 3.55777187437865458875377950042, 3.69928742332344896828782178825, 4.50968305964953760476989565570, 4.77443602414202535048391254339, 5.39039573573711553871805384457, 5.52652317050472845038906012765, 5.92388809734210315233999904073, 6.11576293597130253520701997911, 6.67163897016025209483958688862, 6.87510090786233085536772419350, 7.27106900226214349429361205246, 7.50981064104359983985970421638
