| L(s) = 1 | − 0.751·3-s − 2.43·9-s − 5.27·11-s + 2.43·13-s + 6.12·17-s − 0.441·19-s − 3.18·23-s + 4.08·27-s − 1.25·29-s − 0.645·31-s + 3.96·33-s + 10.7·37-s − 1.82·39-s + 8.90·41-s − 10.4·43-s − 7.52·47-s − 4.60·51-s + 3.34·53-s + 0.331·57-s − 8.15·59-s − 3.47·61-s + 4.78·67-s + 2.39·69-s + 13.7·71-s − 5.32·73-s + 5.40·79-s + 4.23·81-s + ⋯ |
| L(s) = 1 | − 0.433·3-s − 0.811·9-s − 1.59·11-s + 0.675·13-s + 1.48·17-s − 0.101·19-s − 0.664·23-s + 0.786·27-s − 0.232·29-s − 0.115·31-s + 0.690·33-s + 1.76·37-s − 0.293·39-s + 1.39·41-s − 1.58·43-s − 1.09·47-s − 0.644·51-s + 0.458·53-s + 0.0439·57-s − 1.06·59-s − 0.444·61-s + 0.584·67-s + 0.288·69-s + 1.62·71-s − 0.623·73-s + 0.608·79-s + 0.470·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.751T + 3T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 - 2.43T + 13T^{2} \) |
| 17 | \( 1 - 6.12T + 17T^{2} \) |
| 19 | \( 1 + 0.441T + 19T^{2} \) |
| 23 | \( 1 + 3.18T + 23T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 + 0.645T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 8.90T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 7.52T + 47T^{2} \) |
| 53 | \( 1 - 3.34T + 53T^{2} \) |
| 59 | \( 1 + 8.15T + 59T^{2} \) |
| 61 | \( 1 + 3.47T + 61T^{2} \) |
| 67 | \( 1 - 4.78T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 5.32T + 73T^{2} \) |
| 79 | \( 1 - 5.40T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 - 8.60T + 89T^{2} \) |
| 97 | \( 1 + 13.6T + 97T^{2} \) |
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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
−7.52697486873572730979465104928, −6.42957767635694554283434059592, −5.91600329115945951593832754983, −5.35692917399204489042480664653, −4.77619499422663551363821748221, −3.70480245015402052479093130175, −3.01059066556189901171166118575, −2.28112822429062613669692012147, −1.05298804070957094783784030883, 0,
1.05298804070957094783784030883, 2.28112822429062613669692012147, 3.01059066556189901171166118575, 3.70480245015402052479093130175, 4.77619499422663551363821748221, 5.35692917399204489042480664653, 5.91600329115945951593832754983, 6.42957767635694554283434059592, 7.52697486873572730979465104928
