| L(s) = 1 | − 4-s − 3·9-s − 10·11-s + 16-s + 6·19-s + 12·29-s − 8·31-s + 3·36-s + 22·41-s + 10·44-s − 49-s − 8·59-s − 4·61-s − 64-s − 20·71-s − 6·76-s + 4·79-s + 22·89-s + 30·99-s + 36·109-s − 12·116-s + 53·121-s + 8·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 9-s − 3.01·11-s + 1/4·16-s + 1.37·19-s + 2.22·29-s − 1.43·31-s + 1/2·36-s + 3.43·41-s + 1.50·44-s − 1/7·49-s − 1.04·59-s − 0.512·61-s − 1/8·64-s − 2.37·71-s − 0.688·76-s + 0.450·79-s + 2.33·89-s + 3.01·99-s + 3.44·109-s − 1.11·116-s + 4.81·121-s + 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8408229588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8408229588\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−11.71213532525521739404529331675, −11.03248914126200209180492012179, −10.85847408081630898113138468309, −10.35013153478523174428802262470, −9.994343734392532640515735584001, −9.370667495287345675425294175021, −8.921340809713994409888151596341, −8.432901335985605660532466935114, −7.900121616147704622314811869964, −7.51444651436689669043190127183, −7.34160332609263357086643054592, −6.00923260737645566883775136857, −5.95850393584290940415092698715, −5.26795596119065845092726418093, −4.89340268853846218069727323280, −4.35538657252140008829443741148, −3.09017020285620483598189848717, −3.01173303664762003186045602547, −2.25592963732092920140091820731, −0.62531841110652152098686894891,
0.62531841110652152098686894891, 2.25592963732092920140091820731, 3.01173303664762003186045602547, 3.09017020285620483598189848717, 4.35538657252140008829443741148, 4.89340268853846218069727323280, 5.26795596119065845092726418093, 5.95850393584290940415092698715, 6.00923260737645566883775136857, 7.34160332609263357086643054592, 7.51444651436689669043190127183, 7.900121616147704622314811869964, 8.432901335985605660532466935114, 8.921340809713994409888151596341, 9.370667495287345675425294175021, 9.994343734392532640515735584001, 10.35013153478523174428802262470, 10.85847408081630898113138468309, 11.03248914126200209180492012179, 11.71213532525521739404529331675
