| L(s) = 1 | + 3-s − 1.41·5-s + 2.82·7-s + 9-s + 4.24·13-s − 1.41·15-s + 4·17-s + 8·19-s + 2.82·21-s − 5.65·23-s − 2.99·25-s + 27-s + 1.41·29-s − 2.82·31-s − 4.00·35-s + 4.24·37-s + 4.24·39-s − 4·41-s − 1.41·45-s − 11.3·47-s + 1.00·49-s + 4·51-s − 7.07·53-s + 8·57-s + 12·59-s + 1.41·61-s + 2.82·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.632·5-s + 1.06·7-s + 0.333·9-s + 1.17·13-s − 0.365·15-s + 0.970·17-s + 1.83·19-s + 0.617·21-s − 1.17·23-s − 0.599·25-s + 0.192·27-s + 0.262·29-s − 0.508·31-s − 0.676·35-s + 0.697·37-s + 0.679·39-s − 0.624·41-s − 0.210·45-s − 1.65·47-s + 0.142·49-s + 0.560·51-s − 0.971·53-s + 1.05·57-s + 1.56·59-s + 0.181·61-s + 0.356·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.640456034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.640456034\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 + 7.07T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−8.432496650642910447662687843999, −7.926364122584422249886224716653, −7.62910462408589880256070895349, −6.50000445127158845451722435389, −5.54451027429731727038655227701, −4.82108806988171779309151981734, −3.74745215095868329006535082211, −3.35339873067094655780328422771, −1.95695379637772425718145184067, −1.04522955524119973773661302468,
1.04522955524119973773661302468, 1.95695379637772425718145184067, 3.35339873067094655780328422771, 3.74745215095868329006535082211, 4.82108806988171779309151981734, 5.54451027429731727038655227701, 6.50000445127158845451722435389, 7.62910462408589880256070895349, 7.926364122584422249886224716653, 8.432496650642910447662687843999
