| L(s) = 1 | − 3·5-s − 4·7-s + 13-s + 6·17-s + 8·19-s + 5·25-s + 9·29-s − 4·31-s + 12·35-s − 2·37-s + 6·41-s + 8·43-s + 12·47-s + 7·49-s + 12·53-s + 61-s − 3·65-s − 4·67-s − 24·71-s + 22·73-s − 16·79-s + 12·83-s − 18·85-s + 6·89-s − 4·91-s − 24·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.51·7-s + 0.277·13-s + 1.45·17-s + 1.83·19-s + 25-s + 1.67·29-s − 0.718·31-s + 2.02·35-s − 0.328·37-s + 0.937·41-s + 1.21·43-s + 1.75·47-s + 49-s + 1.64·53-s + 0.128·61-s − 0.372·65-s − 0.488·67-s − 2.84·71-s + 2.57·73-s − 1.80·79-s + 1.31·83-s − 1.95·85-s + 0.635·89-s − 0.419·91-s − 2.46·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.609993090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.609993090\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 12 T + 61 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.971464619570892566025350264745, −9.425186132969605128752203026029, −9.013913198769548838075093542767, −8.766467431775290903638156691973, −8.137259291917816043118933199717, −7.77398447808587926281510681957, −7.28185181323512946689732896319, −7.22031512434760449912064748604, −6.72467639090079128148832262360, −5.99649696803467346375902529447, −5.69100725606541307670848786556, −5.45783566444143819342691536576, −4.44449747710178832331294550631, −4.39709510210782604752216641986, −3.45603402797998639510096728836, −3.42929634736446864710308860066, −3.00039605802937088640114894477, −2.30852222462586609344772542945, −0.995591272542316057194106228394, −0.70479586560519622839123601216,
0.70479586560519622839123601216, 0.995591272542316057194106228394, 2.30852222462586609344772542945, 3.00039605802937088640114894477, 3.42929634736446864710308860066, 3.45603402797998639510096728836, 4.39709510210782604752216641986, 4.44449747710178832331294550631, 5.45783566444143819342691536576, 5.69100725606541307670848786556, 5.99649696803467346375902529447, 6.72467639090079128148832262360, 7.22031512434760449912064748604, 7.28185181323512946689732896319, 7.77398447808587926281510681957, 8.137259291917816043118933199717, 8.766467431775290903638156691973, 9.013913198769548838075093542767, 9.425186132969605128752203026029, 9.971464619570892566025350264745
