| L(s) = 1 | + 3-s − 2·5-s − 4.46·7-s − 2·9-s − 4·11-s + 4.46·13-s − 2·15-s − 7.92·17-s + 19-s − 4.46·21-s + 6.46·23-s − 25-s − 5·27-s − 2.46·29-s − 4.92·31-s − 4·33-s + 8.92·35-s − 2·37-s + 4.46·39-s − 6.92·41-s + 4.92·43-s + 4·45-s + 6.92·47-s + 12.9·49-s − 7.92·51-s + 12.4·53-s + 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.68·7-s − 0.666·9-s − 1.20·11-s + 1.23·13-s − 0.516·15-s − 1.92·17-s + 0.229·19-s − 0.974·21-s + 1.34·23-s − 0.200·25-s − 0.962·27-s − 0.457·29-s − 0.885·31-s − 0.696·33-s + 1.50·35-s − 0.328·37-s + 0.714·39-s − 1.08·41-s + 0.751·43-s + 0.596·45-s + 1.01·47-s + 1.84·49-s − 1.11·51-s + 1.71·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5989487489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5989487489\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 + 4.46T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4.46T + 13T^{2} \) |
| 17 | \( 1 + 7.92T + 17T^{2} \) |
| 23 | \( 1 - 6.46T + 23T^{2} \) |
| 29 | \( 1 + 2.46T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 9.92T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 5.92T + 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 2.92T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−8.409844142482034624542865921103, −7.51076617999811505742478471969, −6.95558838694580589641188612609, −6.13451387595584215331300907883, −5.46671440922679715753527619258, −4.32028731814252122288459318747, −3.51893930683257661830686548440, −3.06474870964278006041252469469, −2.21907037782180396589358294394, −0.38113433795143368092123168877,
0.38113433795143368092123168877, 2.21907037782180396589358294394, 3.06474870964278006041252469469, 3.51893930683257661830686548440, 4.32028731814252122288459318747, 5.46671440922679715753527619258, 6.13451387595584215331300907883, 6.95558838694580589641188612609, 7.51076617999811505742478471969, 8.409844142482034624542865921103
