L(s) = 1 | − 3-s − 5-s + 4.21·7-s + 9-s − 5.57·11-s − 2.21·13-s + 15-s + 0.643·17-s + 19-s − 4.21·21-s − 1.35·23-s + 25-s − 27-s + 4.86·29-s − 2.64·31-s + 5.57·33-s − 4.21·35-s − 2.21·37-s + 2.21·39-s + 3.57·41-s + 0.218·43-s − 45-s + 1.35·47-s + 10.7·49-s − 0.643·51-s + 0.643·53-s + 5.57·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.59·7-s + 0.333·9-s − 1.68·11-s − 0.615·13-s + 0.258·15-s + 0.156·17-s + 0.229·19-s − 0.920·21-s − 0.282·23-s + 0.200·25-s − 0.192·27-s + 0.902·29-s − 0.474·31-s + 0.970·33-s − 0.713·35-s − 0.364·37-s + 0.355·39-s + 0.558·41-s + 0.0332·43-s − 0.149·45-s + 0.197·47-s + 1.54·49-s − 0.0900·51-s + 0.0883·53-s + 0.751·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.353088184\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.353088184\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 4.21T + 7T^{2} \) |
| 11 | \( 1 + 5.57T + 11T^{2} \) |
| 13 | \( 1 + 2.21T + 13T^{2} \) |
| 17 | \( 1 - 0.643T + 17T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 - 4.86T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 + 2.21T + 37T^{2} \) |
| 41 | \( 1 - 3.57T + 41T^{2} \) |
| 43 | \( 1 - 0.218T + 43T^{2} \) |
| 47 | \( 1 - 1.35T + 47T^{2} \) |
| 53 | \( 1 - 0.643T + 53T^{2} \) |
| 59 | \( 1 - 4.43T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 1.28T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 1.28T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 5.78T + 97T^{2} \) |
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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
−8.174665113911723343324402009095, −7.60829909035508706332576000707, −7.14895817382775261480952973757, −5.95015214540643568874222890407, −5.12633724426551357687722013577, −4.90857428746292522172906084883, −3.99384385244426311562481846001, −2.75854425724953520010649723259, −1.91836667308977616396531829929, −0.66301121612278665301158155634,
0.66301121612278665301158155634, 1.91836667308977616396531829929, 2.75854425724953520010649723259, 3.99384385244426311562481846001, 4.90857428746292522172906084883, 5.12633724426551357687722013577, 5.95015214540643568874222890407, 7.14895817382775261480952973757, 7.60829909035508706332576000707, 8.174665113911723343324402009095