| L(s) = 1 | − 3-s + 5-s + 9-s + 4·11-s + 6·13-s − 15-s + 2·17-s − 4·19-s + 8·23-s + 25-s − 27-s + 6·29-s + 31-s − 4·33-s − 2·37-s − 6·39-s + 10·41-s + 4·43-s + 45-s − 7·49-s − 2·51-s − 10·53-s + 4·55-s + 4·57-s + 12·59-s − 2·61-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.179·31-s − 0.696·33-s − 0.328·37-s − 0.960·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 49-s − 0.280·51-s − 1.37·53-s + 0.539·55-s + 0.529·57-s + 1.56·59-s − 0.256·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.488400491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488400491\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.952418916075434749361279985637, −6.88246346082924064334153856038, −6.46786428019455517543118893027, −5.96454234827988216942923974692, −5.14595438983531183116874464717, −4.31899860893846250195620811503, −3.66329218384675427078549218754, −2.73098877920014273376825766211, −1.46104309889883464380034770543, −0.945501887428634190883132606774,
0.945501887428634190883132606774, 1.46104309889883464380034770543, 2.73098877920014273376825766211, 3.66329218384675427078549218754, 4.31899860893846250195620811503, 5.14595438983531183116874464717, 5.96454234827988216942923974692, 6.46786428019455517543118893027, 6.88246346082924064334153856038, 7.952418916075434749361279985637
