| L(s) = 1 | − 3-s − 5·7-s + 2·11-s − 2·13-s − 4·17-s + 19-s + 5·21-s + 4·23-s + 27-s + 5·31-s − 2·33-s − 5·37-s + 2·39-s + 4·41-s + 18·43-s − 2·47-s + 18·49-s + 4·51-s + 12·53-s − 57-s + 8·59-s + 14·61-s + 9·67-s − 4·69-s + 4·71-s + 73-s − 10·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s + 0.603·11-s − 0.554·13-s − 0.970·17-s + 0.229·19-s + 1.09·21-s + 0.834·23-s + 0.192·27-s + 0.898·31-s − 0.348·33-s − 0.821·37-s + 0.320·39-s + 0.624·41-s + 2.74·43-s − 0.291·47-s + 18/7·49-s + 0.560·51-s + 1.64·53-s − 0.132·57-s + 1.04·59-s + 1.79·61-s + 1.09·67-s − 0.481·69-s + 0.474·71-s + 0.117·73-s − 1.13·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.555048982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.555048982\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 8 T + 5 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.262038368923203981098632329300, −9.132300554074104576702776868188, −8.482758086048450151399171804006, −8.375183588739223037009572822959, −7.50172183837169471511247503397, −7.23216733229752729036010198526, −6.85450940069697756335436256361, −6.64476525534873025548933531443, −6.06202521146175299630536655540, −5.99025644791708797124711972745, −5.19147639734038285780746183860, −5.14266093675108710307100003395, −4.28774932010403334985069544584, −4.00659228161116918972442654404, −3.56046602908367554408343595623, −3.03137428152345908397430881329, −2.33458275273852578141090617378, −2.27737568507713377904885638351, −0.77444188146302283216486704328, −0.69536828672219612288057335657,
0.69536828672219612288057335657, 0.77444188146302283216486704328, 2.27737568507713377904885638351, 2.33458275273852578141090617378, 3.03137428152345908397430881329, 3.56046602908367554408343595623, 4.00659228161116918972442654404, 4.28774932010403334985069544584, 5.14266093675108710307100003395, 5.19147639734038285780746183860, 5.99025644791708797124711972745, 6.06202521146175299630536655540, 6.64476525534873025548933531443, 6.85450940069697756335436256361, 7.23216733229752729036010198526, 7.50172183837169471511247503397, 8.375183588739223037009572822959, 8.482758086048450151399171804006, 9.132300554074104576702776868188, 9.262038368923203981098632329300
