| L(s) = 1 | − 1.72·3-s + 0.586·5-s − 2.46·7-s − 0.0259·9-s + 2.73·11-s + 1.16·13-s − 1.01·15-s − 3.49·17-s + 5.19·19-s + 4.24·21-s − 1.89·23-s − 4.65·25-s + 5.21·27-s − 3.57·29-s − 1.26·31-s − 4.70·33-s − 1.44·35-s + 10.5·37-s − 2.00·39-s − 4.19·41-s + 8.57·43-s − 0.0152·45-s + 4.29·47-s − 0.944·49-s + 6.02·51-s − 6.91·53-s + 1.60·55-s + ⋯ |
| L(s) = 1 | − 0.995·3-s + 0.262·5-s − 0.930·7-s − 0.00864·9-s + 0.823·11-s + 0.322·13-s − 0.261·15-s − 0.847·17-s + 1.19·19-s + 0.926·21-s − 0.395·23-s − 0.931·25-s + 1.00·27-s − 0.663·29-s − 0.226·31-s − 0.819·33-s − 0.243·35-s + 1.73·37-s − 0.321·39-s − 0.654·41-s + 1.30·43-s − 0.00226·45-s + 0.626·47-s − 0.134·49-s + 0.843·51-s − 0.949·53-s + 0.215·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 1.72T + 3T^{2} \) |
| 5 | \( 1 - 0.586T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 + 3.57T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 4.19T + 41T^{2} \) |
| 43 | \( 1 - 8.57T + 43T^{2} \) |
| 47 | \( 1 - 4.29T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 + 6.43T + 59T^{2} \) |
| 61 | \( 1 - 5.28T + 61T^{2} \) |
| 67 | \( 1 - 7.31T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 + 5.43T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 + 3.03T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−7.988151891599409833716038180681, −7.17100806358713176985513613165, −6.29149010715932160802301398699, −6.07353723148317124614491159917, −5.27349245581549318411506921030, −4.28399489035673408216582393134, −3.50430066307361477340797235121, −2.47706244087303742043079160970, −1.18740090280235619050599323004, 0,
1.18740090280235619050599323004, 2.47706244087303742043079160970, 3.50430066307361477340797235121, 4.28399489035673408216582393134, 5.27349245581549318411506921030, 6.07353723148317124614491159917, 6.29149010715932160802301398699, 7.17100806358713176985513613165, 7.988151891599409833716038180681
