| L(s) = 1 | − 3·9-s + 10·11-s − 4·19-s − 14·29-s − 8·31-s − 24·41-s − 49-s − 8·59-s + 8·61-s − 6·79-s − 30·99-s − 36·101-s − 10·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 12·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 9-s + 3.01·11-s − 0.917·19-s − 2.59·29-s − 1.43·31-s − 3.74·41-s − 1/7·49-s − 1.04·59-s + 1.02·61-s − 0.675·79-s − 3.01·99-s − 3.58·101-s − 0.957·109-s + 4.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 0.917·171-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.195598518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.195598518\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 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 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−9.331243751693377777695890021028, −8.588781150376436725579490489294, −8.260569503153250233009830913617, −8.139583181279148725646084066985, −7.28323647905176075416117736055, −6.92881110672467810626490745442, −6.79461828124479501283005807905, −6.46918025374441034590737957831, −5.84709066790486068284477349870, −5.48625451372435419391067603093, −5.42361587279815601354953828351, −4.36903042101480602545038260618, −4.36241471687044704903841453917, −3.72571509984016031491542950728, −3.41272955473304130281038843477, −3.15789904145046700917368455608, −2.13466988899724902419132066078, −1.65617571331981277750848083600, −1.54626619684282501890904015405, −0.34047552422091485053748234609,
0.34047552422091485053748234609, 1.54626619684282501890904015405, 1.65617571331981277750848083600, 2.13466988899724902419132066078, 3.15789904145046700917368455608, 3.41272955473304130281038843477, 3.72571509984016031491542950728, 4.36241471687044704903841453917, 4.36903042101480602545038260618, 5.42361587279815601354953828351, 5.48625451372435419391067603093, 5.84709066790486068284477349870, 6.46918025374441034590737957831, 6.79461828124479501283005807905, 6.92881110672467810626490745442, 7.28323647905176075416117736055, 8.139583181279148725646084066985, 8.260569503153250233009830913617, 8.588781150376436725579490489294, 9.331243751693377777695890021028
