L(s) = 1 | + 2·3-s + 5-s + 9-s + 4·11-s + 4·13-s + 2·15-s − 2·17-s − 19-s + 4·23-s + 25-s − 4·27-s − 6·29-s + 8·31-s + 8·33-s − 4·37-s + 8·39-s − 2·41-s − 4·43-s + 45-s + 8·47-s − 7·49-s − 4·51-s + 4·55-s − 2·57-s + 8·59-s + 2·61-s + 4·65-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.10·13-s + 0.516·15-s − 0.485·17-s − 0.229·19-s + 0.834·23-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 1.43·31-s + 1.39·33-s − 0.657·37-s + 1.28·39-s − 0.312·41-s − 0.609·43-s + 0.149·45-s + 1.16·47-s − 49-s − 0.560·51-s + 0.539·55-s − 0.264·57-s + 1.04·59-s + 0.256·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.966359563\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.966359563\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−9.293388595910869687909126342500, −8.676622547521425975649264857286, −8.201244304636770949086799479495, −6.98504233521847162657380496703, −6.38944943987389263321888989842, −5.36201935903527945958183748631, −4.09827981349116155846681566097, −3.44338427084671849598446043190, −2.37506071445705739411737524146, −1.32637780105615946086112532423,
1.32637780105615946086112532423, 2.37506071445705739411737524146, 3.44338427084671849598446043190, 4.09827981349116155846681566097, 5.36201935903527945958183748631, 6.38944943987389263321888989842, 6.98504233521847162657380496703, 8.201244304636770949086799479495, 8.676622547521425975649264857286, 9.293388595910869687909126342500