| L(s) = 1 | − 9-s − 4·11-s − 10·19-s − 20·29-s + 6·31-s − 16·41-s + 5·49-s − 20·59-s + 14·61-s + 16·71-s + 81-s + 4·99-s + 24·101-s − 10·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 10·171-s + 173-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.20·11-s − 2.29·19-s − 3.71·29-s + 1.07·31-s − 2.49·41-s + 5/7·49-s − 2.60·59-s + 1.79·61-s + 1.89·71-s + 1/9·81-s + 0.402·99-s + 2.38·101-s − 0.957·109-s − 0.909·121-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 + 1.92·169-s + 0.764·171-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4917795709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4917795709\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−9.949907055488745137097779037514, −9.578495447981886821356180020888, −9.064836626535550633354138233983, −8.716295803944429446485180625998, −8.340335244204874851631296247914, −7.86234145614925000217120166161, −7.66888427676943697701842596601, −7.02800850792565103921338730538, −6.64656603942778753571332266857, −6.16619618781521749947669039417, −5.77248322643970656018602756557, −5.19405509753699174454130377229, −4.99944718464177753990442637794, −4.29625234040689046689858087905, −3.78438288556302106727629890536, −3.40900263759224670531452537491, −2.63293869948519873765064966244, −2.10770564300735768058513110529, −1.72634591722671636220940902706, −0.28079064034016865538553634585,
0.28079064034016865538553634585, 1.72634591722671636220940902706, 2.10770564300735768058513110529, 2.63293869948519873765064966244, 3.40900263759224670531452537491, 3.78438288556302106727629890536, 4.29625234040689046689858087905, 4.99944718464177753990442637794, 5.19405509753699174454130377229, 5.77248322643970656018602756557, 6.16619618781521749947669039417, 6.64656603942778753571332266857, 7.02800850792565103921338730538, 7.66888427676943697701842596601, 7.86234145614925000217120166161, 8.340335244204874851631296247914, 8.716295803944429446485180625998, 9.064836626535550633354138233983, 9.578495447981886821356180020888, 9.949907055488745137097779037514
