| L(s) = 1 | + 5·9-s − 8·11-s + 8·19-s + 14·29-s − 14·31-s + 6·41-s + 10·49-s + 16·59-s + 26·71-s − 8·79-s + 16·81-s + 12·89-s − 40·99-s + 20·101-s − 20·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 40·171-s + ⋯ |
| L(s) = 1 | + 5/3·9-s − 2.41·11-s + 1.83·19-s + 2.59·29-s − 2.51·31-s + 0.937·41-s + 10/7·49-s + 2.08·59-s + 3.08·71-s − 0.900·79-s + 16/9·81-s + 1.27·89-s − 4.02·99-s + 1.99·101-s − 1.91·109-s + 2.36·121-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 + 3.05·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.811802161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.811802161\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 75 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.074625954016430175801268524195, −9.030005885000182364250671179469, −8.124952866548397436076861021061, −8.092996592595659629131241977481, −7.62414823643312068949768306731, −7.36479028653441008341503796377, −6.86301225767045164769790908098, −6.79178784176427831827459873090, −5.96502927929094857478122369389, −5.49339557191206316639794891975, −5.16422828327927974436411089444, −5.06011110946550710118230472623, −4.34874307326951855601110891879, −4.02042298853308908920775094412, −3.36069239427699225772526922107, −3.02962866390811536115653112262, −2.27960170628735718837999909547, −2.12925851821850225768871160118, −1.09345305363549017705766259297, −0.67480558893572854577453331746,
0.67480558893572854577453331746, 1.09345305363549017705766259297, 2.12925851821850225768871160118, 2.27960170628735718837999909547, 3.02962866390811536115653112262, 3.36069239427699225772526922107, 4.02042298853308908920775094412, 4.34874307326951855601110891879, 5.06011110946550710118230472623, 5.16422828327927974436411089444, 5.49339557191206316639794891975, 5.96502927929094857478122369389, 6.79178784176427831827459873090, 6.86301225767045164769790908098, 7.36479028653441008341503796377, 7.62414823643312068949768306731, 8.092996592595659629131241977481, 8.124952866548397436076861021061, 9.030005885000182364250671179469, 9.074625954016430175801268524195
