| L(s) = 1 | + 2.56·5-s + 3·7-s − 0.561·11-s − 1.56·13-s − 0.123·17-s + 19-s − 5.56·23-s + 1.56·25-s − 4.68·29-s + 9.12·31-s + 7.68·35-s + 7.12·37-s − 4·41-s + 5.43·43-s + 3.68·47-s + 2·49-s + 4.43·53-s − 1.43·55-s + 4.43·59-s + 14.8·61-s − 4·65-s − 0.438·67-s + 4.24·71-s − 9.24·73-s − 1.68·77-s + 10·79-s + 17.3·83-s + ⋯ |
| L(s) = 1 | + 1.14·5-s + 1.13·7-s − 0.169·11-s − 0.433·13-s − 0.0298·17-s + 0.229·19-s − 1.15·23-s + 0.312·25-s − 0.869·29-s + 1.63·31-s + 1.29·35-s + 1.17·37-s − 0.624·41-s + 0.829·43-s + 0.537·47-s + 0.285·49-s + 0.609·53-s − 0.193·55-s + 0.577·59-s + 1.89·61-s − 0.496·65-s − 0.0535·67-s + 0.503·71-s − 1.08·73-s − 0.191·77-s + 1.12·79-s + 1.90·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.888061977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.888061977\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 3T + 7T^{2} \) |
| 11 | \( 1 + 0.561T + 11T^{2} \) |
| 13 | \( 1 + 1.56T + 13T^{2} \) |
| 17 | \( 1 + 0.123T + 17T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 + 4.68T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 - 3.68T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 4.43T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 + 0.438T + 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 9.24T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−8.042375524290343568199281642834, −7.64163850794416004344796674669, −6.62081544760676749321757023340, −5.93744203980919319427482091975, −5.30073930920283809331793726000, −4.64820783072911342871159965307, −3.77855011782730182528548303337, −2.44895219823249263787187684678, −2.04333652287247103430469472446, −0.942066293094218888838118614618,
0.942066293094218888838118614618, 2.04333652287247103430469472446, 2.44895219823249263787187684678, 3.77855011782730182528548303337, 4.64820783072911342871159965307, 5.30073930920283809331793726000, 5.93744203980919319427482091975, 6.62081544760676749321757023340, 7.64163850794416004344796674669, 8.042375524290343568199281642834
