| L(s) = 1 | − 4-s + 2·5-s − 3·9-s + 16-s + 4·19-s − 2·20-s − 25-s + 2·29-s + 20·31-s + 3·36-s − 6·41-s − 6·45-s − 4·59-s − 18·61-s − 64-s + 12·71-s − 4·76-s + 20·79-s + 2·80-s + 14·89-s + 8·95-s + 100-s − 30·101-s − 10·109-s − 2·116-s − 22·121-s − 20·124-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.894·5-s − 9-s + 1/4·16-s + 0.917·19-s − 0.447·20-s − 1/5·25-s + 0.371·29-s + 3.59·31-s + 1/2·36-s − 0.937·41-s − 0.894·45-s − 0.520·59-s − 2.30·61-s − 1/8·64-s + 1.42·71-s − 0.458·76-s + 2.25·79-s + 0.223·80-s + 1.48·89-s + 0.820·95-s + 1/10·100-s − 2.98·101-s − 0.957·109-s − 0.185·116-s − 2·121-s − 1.79·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.662250009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.662250009\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−11.10335176679416625910287629110, −10.52799108299470233349145219267, −10.42537526706707692909093735908, −9.588244994212059839987363044695, −9.551227380424728540786670719027, −9.138876349627287243003238478639, −8.367194758357322186195821281066, −8.062432627311967270911524680753, −7.910596101857662333415547699128, −6.73201209166084878866102250758, −6.70655527128002391976556045619, −5.84316179425570037473660220470, −5.78297982913715660147830681699, −4.86840780284918785768322894766, −4.76790435675090586765282056902, −3.89369508975654472232376329879, −3.01173407665259824784727448161, −2.81010372209694162803729543525, −1.82258075705057050787657422887, −0.817136791546388367289942731086,
0.817136791546388367289942731086, 1.82258075705057050787657422887, 2.81010372209694162803729543525, 3.01173407665259824784727448161, 3.89369508975654472232376329879, 4.76790435675090586765282056902, 4.86840780284918785768322894766, 5.78297982913715660147830681699, 5.84316179425570037473660220470, 6.70655527128002391976556045619, 6.73201209166084878866102250758, 7.910596101857662333415547699128, 8.062432627311967270911524680753, 8.367194758357322186195821281066, 9.138876349627287243003238478639, 9.551227380424728540786670719027, 9.588244994212059839987363044695, 10.42537526706707692909093735908, 10.52799108299470233349145219267, 11.10335176679416625910287629110
