| L(s) = 1 | − 2·3-s + 4·7-s + 9-s + 6·13-s − 2·17-s + 4·19-s − 8·21-s − 6·23-s + 4·27-s + 2·29-s − 8·31-s − 8·37-s − 12·39-s − 6·41-s + 12·43-s − 10·47-s + 9·49-s + 4·51-s − 8·57-s + 4·59-s + 10·61-s + 4·63-s − 2·67-s + 12·69-s + 8·71-s − 2·73-s + 4·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.66·13-s − 0.485·17-s + 0.917·19-s − 1.74·21-s − 1.25·23-s + 0.769·27-s + 0.371·29-s − 1.43·31-s − 1.31·37-s − 1.92·39-s − 0.937·41-s + 1.82·43-s − 1.45·47-s + 9/7·49-s + 0.560·51-s − 1.05·57-s + 0.520·59-s + 1.28·61-s + 0.503·63-s − 0.244·67-s + 1.44·69-s + 0.949·71-s − 0.234·73-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.67984312922153, −14.31952738201267, −13.75982194018141, −13.42522623877283, −12.49962291912741, −12.21828265388146, −11.45734917492578, −11.26860667097157, −10.94781651400812, −10.36895735519224, −9.740701112182452, −8.922295227873323, −8.398846768431818, −8.138381947412341, −7.293597260629376, −6.793926107160209, −6.095839683794131, −5.576667284781694, −5.256727863630628, −4.585685712509954, −3.936607395223848, −3.377739158504317, −2.248376331983553, −1.558699911995355, −1.029683553162149, 0,
1.029683553162149, 1.558699911995355, 2.248376331983553, 3.377739158504317, 3.936607395223848, 4.585685712509954, 5.256727863630628, 5.576667284781694, 6.095839683794131, 6.793926107160209, 7.293597260629376, 8.138381947412341, 8.398846768431818, 8.922295227873323, 9.740701112182452, 10.36895735519224, 10.94781651400812, 11.26860667097157, 11.45734917492578, 12.21828265388146, 12.49962291912741, 13.42522623877283, 13.75982194018141, 14.31952738201267, 14.67984312922153
