| L(s) = 1 | − 2·5-s + 4·7-s − 2·13-s + 4·17-s − 4·19-s − 4·23-s + 3·25-s − 4·29-s − 8·35-s + 4·37-s − 4·41-s − 8·47-s − 2·49-s − 8·59-s − 4·61-s + 4·65-s + 16·67-s − 16·73-s + 16·79-s − 16·83-s − 8·85-s − 4·89-s − 8·91-s + 8·95-s − 28·101-s + 16·103-s − 16·107-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.51·7-s − 0.554·13-s + 0.970·17-s − 0.917·19-s − 0.834·23-s + 3/5·25-s − 0.742·29-s − 1.35·35-s + 0.657·37-s − 0.624·41-s − 1.16·47-s − 2/7·49-s − 1.04·59-s − 0.512·61-s + 0.496·65-s + 1.95·67-s − 1.87·73-s + 1.80·79-s − 1.75·83-s − 0.867·85-s − 0.423·89-s − 0.838·91-s + 0.820·95-s − 2.78·101-s + 1.57·103-s − 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87609600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 178 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 16 T + 142 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 174 T^{2} + p^{2} T^{4} \) |
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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
−7.53928565741736582862105781192, −7.36760705411490076301331171588, −6.90763472019682130761682656765, −6.60909506464236844769008738881, −6.11732228827021399748291447723, −5.85721328381193558326302688943, −5.24335244464295165360164816652, −5.20940502283961275252602590700, −4.70302288921122313140494575231, −4.39109677054041065407648951287, −4.13868200220963801418778627032, −3.69289030780213072854720058025, −3.24656157079255885764273954366, −2.93340971749375441862981186755, −2.22369602642011174801552817016, −2.04776720337917005438234830180, −1.30646498703490264752802920334, −1.23372236321440538539139824725, 0, 0,
1.23372236321440538539139824725, 1.30646498703490264752802920334, 2.04776720337917005438234830180, 2.22369602642011174801552817016, 2.93340971749375441862981186755, 3.24656157079255885764273954366, 3.69289030780213072854720058025, 4.13868200220963801418778627032, 4.39109677054041065407648951287, 4.70302288921122313140494575231, 5.20940502283961275252602590700, 5.24335244464295165360164816652, 5.85721328381193558326302688943, 6.11732228827021399748291447723, 6.60909506464236844769008738881, 6.90763472019682130761682656765, 7.36760705411490076301331171588, 7.53928565741736582862105781192
