| L(s) = 1 | + 3.15·3-s + 1.86·5-s + 4.61·7-s + 6.94·9-s − 11-s + 0.224·13-s + 5.86·15-s + 0.675·17-s + 2.35·19-s + 14.5·21-s − 4.72·23-s − 1.54·25-s + 12.4·27-s − 2.15·29-s + 7.11·31-s − 3.15·33-s + 8.58·35-s + 2.53·37-s + 0.706·39-s − 2.11·41-s − 7.05·43-s + 12.9·45-s − 12.1·47-s + 14.3·49-s + 2.13·51-s + 53-s − 1.86·55-s + ⋯ |
| L(s) = 1 | + 1.82·3-s + 0.831·5-s + 1.74·7-s + 2.31·9-s − 0.301·11-s + 0.0621·13-s + 1.51·15-s + 0.163·17-s + 0.539·19-s + 3.17·21-s − 0.984·23-s − 0.308·25-s + 2.39·27-s − 0.399·29-s + 1.27·31-s − 0.548·33-s + 1.45·35-s + 0.416·37-s + 0.113·39-s − 0.330·41-s − 1.07·43-s + 1.92·45-s − 1.77·47-s + 2.04·49-s + 0.298·51-s + 0.137·53-s − 0.250·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.663009669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.663009669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 3.15T + 3T^{2} \) |
| 5 | \( 1 - 1.86T + 5T^{2} \) |
| 7 | \( 1 - 4.61T + 7T^{2} \) |
| 13 | \( 1 - 0.224T + 13T^{2} \) |
| 17 | \( 1 - 0.675T + 17T^{2} \) |
| 19 | \( 1 - 2.35T + 19T^{2} \) |
| 23 | \( 1 + 4.72T + 23T^{2} \) |
| 29 | \( 1 + 2.15T + 29T^{2} \) |
| 31 | \( 1 - 7.11T + 31T^{2} \) |
| 37 | \( 1 - 2.53T + 37T^{2} \) |
| 41 | \( 1 + 2.11T + 41T^{2} \) |
| 43 | \( 1 + 7.05T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 5.17T + 61T^{2} \) |
| 67 | \( 1 + 15.9T + 67T^{2} \) |
| 71 | \( 1 - 3.14T + 71T^{2} \) |
| 73 | \( 1 - 9.88T + 73T^{2} \) |
| 79 | \( 1 - 2.66T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 - 4.78T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−8.014957066267600729476406900137, −7.35647938904285111255569425055, −6.47367135555996248837813569309, −5.50501018217347575058130798292, −4.81590214794307271031728677112, −4.16174861070577838236168911903, −3.31140704289394879640529828447, −2.45878198874396957106661229687, −1.86736275388179524090044406612, −1.31927649220957758423924726126,
1.31927649220957758423924726126, 1.86736275388179524090044406612, 2.45878198874396957106661229687, 3.31140704289394879640529828447, 4.16174861070577838236168911903, 4.81590214794307271031728677112, 5.50501018217347575058130798292, 6.47367135555996248837813569309, 7.35647938904285111255569425055, 8.014957066267600729476406900137
