| L(s) = 1 | + 3·3-s − 5·5-s + 7·7-s + 9·9-s + 33.6·11-s − 38.3·13-s − 15·15-s + 65.7·17-s − 33.3·19-s + 21·21-s − 207.·23-s + 25·25-s + 27·27-s − 189.·29-s − 202.·31-s + 100.·33-s − 35·35-s − 16.5·37-s − 115.·39-s + 388.·41-s − 41.8·43-s − 45·45-s − 368.·47-s + 49·49-s + 197.·51-s + 458.·53-s − 168.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.921·11-s − 0.818·13-s − 0.258·15-s + 0.937·17-s − 0.403·19-s + 0.218·21-s − 1.88·23-s + 0.200·25-s + 0.192·27-s − 1.21·29-s − 1.17·31-s + 0.531·33-s − 0.169·35-s − 0.0734·37-s − 0.472·39-s + 1.48·41-s − 0.148·43-s − 0.149·45-s − 1.14·47-s + 0.142·49-s + 0.541·51-s + 1.18·53-s − 0.412·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 - 7T \) |
| good | 11 | \( 1 - 33.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 65.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 207.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 189.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 16.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 388.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 41.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 368.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 458.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 256.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 123.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 336.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 22.0T + 3.89e5T^{2} \) |
| 79 | \( 1 + 385.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 23.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 51.9T + 9.12e5T^{2} \) |
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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
−8.531217576226349885976973369650, −7.76457245653019079713027999377, −7.27875188619155206348332621163, −6.19315639728340030314052038914, −5.27784427751853549606884065647, −4.13963231645660678809508226152, −3.66914952388421824495906849099, −2.38715737553529385590759364152, −1.45988546523222256414934791375, 0,
1.45988546523222256414934791375, 2.38715737553529385590759364152, 3.66914952388421824495906849099, 4.13963231645660678809508226152, 5.27784427751853549606884065647, 6.19315639728340030314052038914, 7.27875188619155206348332621163, 7.76457245653019079713027999377, 8.531217576226349885976973369650
