| L(s) = 1 | − 3-s + 3.34·5-s − 3.34·7-s + 9-s − 0.402·11-s + 1.84·13-s − 3.34·15-s + 1.59·17-s + 1.34·19-s + 3.34·21-s + 5.34·23-s + 6.19·25-s − 27-s + 3.59·29-s + 5.44·31-s + 0.402·33-s − 11.1·35-s − 3.44·37-s − 1.84·39-s − 41-s − 5.24·43-s + 3.34·45-s + 5.59·47-s + 4.19·49-s − 1.59·51-s + 4·53-s − 1.34·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.49·5-s − 1.26·7-s + 0.333·9-s − 0.121·11-s + 0.512·13-s − 0.863·15-s + 0.387·17-s + 0.308·19-s + 0.730·21-s + 1.11·23-s + 1.23·25-s − 0.192·27-s + 0.668·29-s + 0.978·31-s + 0.0700·33-s − 1.89·35-s − 0.566·37-s − 0.296·39-s − 0.156·41-s − 0.799·43-s + 0.498·45-s + 0.816·47-s + 0.599·49-s − 0.223·51-s + 0.549·53-s − 0.181·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.053294267\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.053294267\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| good | 5 | \( 1 - 3.34T + 5T^{2} \) |
| 7 | \( 1 + 3.34T + 7T^{2} \) |
| 11 | \( 1 + 0.402T + 11T^{2} \) |
| 13 | \( 1 - 1.84T + 13T^{2} \) |
| 17 | \( 1 - 1.59T + 17T^{2} \) |
| 19 | \( 1 - 1.34T + 19T^{2} \) |
| 23 | \( 1 - 5.34T + 23T^{2} \) |
| 29 | \( 1 - 3.59T + 29T^{2} \) |
| 31 | \( 1 - 5.44T + 31T^{2} \) |
| 37 | \( 1 + 3.44T + 37T^{2} \) |
| 43 | \( 1 + 5.24T + 43T^{2} \) |
| 47 | \( 1 - 5.59T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 14.6T + 59T^{2} \) |
| 61 | \( 1 - 7.44T + 61T^{2} \) |
| 67 | \( 1 + 7.19T + 67T^{2} \) |
| 71 | \( 1 - 9.09T + 71T^{2} \) |
| 73 | \( 1 + 3.44T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 + 6.15T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 + 2.54T + 97T^{2} \) |
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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
−7.73673667865337801632210744170, −6.68486542980477085336635483025, −6.57894320956853230752690403420, −5.75927418148115212770047079204, −5.32357772020692907655568321706, −4.44566790277674810322131828152, −3.30081673104774022272240038934, −2.77081390583119132972413283466, −1.66975118829179782240778148839, −0.75707395730893971177198773661,
0.75707395730893971177198773661, 1.66975118829179782240778148839, 2.77081390583119132972413283466, 3.30081673104774022272240038934, 4.44566790277674810322131828152, 5.32357772020692907655568321706, 5.75927418148115212770047079204, 6.57894320956853230752690403420, 6.68486542980477085336635483025, 7.73673667865337801632210744170
