| L(s) = 1 | − 3-s + 2·7-s + 9-s + 6·11-s − 2·13-s − 2·21-s + 23-s − 27-s − 6·29-s + 8·31-s − 6·33-s + 2·39-s + 10·41-s − 12·43-s + 8·47-s − 3·49-s + 2·53-s + 12·59-s − 4·61-s + 2·63-s − 12·67-s − 69-s + 10·73-s + 12·77-s − 6·79-s + 81-s + 14·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s − 0.436·21-s + 0.208·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 1.04·33-s + 0.320·39-s + 1.56·41-s − 1.82·43-s + 1.16·47-s − 3/7·49-s + 0.274·53-s + 1.56·59-s − 0.512·61-s + 0.251·63-s − 1.46·67-s − 0.120·69-s + 1.17·73-s + 1.36·77-s − 0.675·79-s + 1/9·81-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.819714615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.819714615\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.75212536438109, −13.12652263777677, −12.57528262823170, −12.02419667467641, −11.67087876395119, −11.42127406538706, −10.80568563193393, −10.28421311471135, −9.644325905891330, −9.349152354186453, −8.693587270850216, −8.282930944753553, −7.497404539417170, −7.204141353615567, −6.542154459594482, −6.125400630746189, −5.593595671050876, −4.813846476530998, −4.574164092505491, −3.888903198295876, −3.404894143205962, −2.456013504505726, −1.830241428422378, −1.196543804812100, −0.5970081553704622,
0.5970081553704622, 1.196543804812100, 1.830241428422378, 2.456013504505726, 3.404894143205962, 3.888903198295876, 4.574164092505491, 4.813846476530998, 5.593595671050876, 6.125400630746189, 6.542154459594482, 7.204141353615567, 7.497404539417170, 8.282930944753553, 8.693587270850216, 9.349152354186453, 9.644325905891330, 10.28421311471135, 10.80568563193393, 11.42127406538706, 11.67087876395119, 12.02419667467641, 12.57528262823170, 13.12652263777677, 13.75212536438109
