L(s) = 1 | − 1.61·3-s − 0.419·5-s + 5.20·7-s − 0.405·9-s − 5.36·11-s + 6.78·13-s + 0.676·15-s + 2.80·17-s + 6.53·19-s − 8.38·21-s + 4.71·23-s − 4.82·25-s + 5.48·27-s + 8.91·29-s − 2.32·31-s + 8.63·33-s − 2.18·35-s + 10.2·37-s − 10.9·39-s − 7.34·41-s + 1.47·43-s + 0.169·45-s − 2.41·47-s + 20.1·49-s − 4.51·51-s − 2.63·53-s + 2.25·55-s + ⋯ |
L(s) = 1 | − 0.930·3-s − 0.187·5-s + 1.96·7-s − 0.135·9-s − 1.61·11-s + 1.88·13-s + 0.174·15-s + 0.679·17-s + 1.49·19-s − 1.83·21-s + 0.982·23-s − 0.964·25-s + 1.05·27-s + 1.65·29-s − 0.417·31-s + 1.50·33-s − 0.369·35-s + 1.68·37-s − 1.75·39-s − 1.14·41-s + 0.224·43-s + 0.0253·45-s − 0.352·47-s + 2.87·49-s − 0.631·51-s − 0.361·53-s + 0.303·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.005822448\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.005822448\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + 1.61T + 3T^{2} \) |
| 5 | \( 1 + 0.419T + 5T^{2} \) |
| 7 | \( 1 - 5.20T + 7T^{2} \) |
| 11 | \( 1 + 5.36T + 11T^{2} \) |
| 13 | \( 1 - 6.78T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 19 | \( 1 - 6.53T + 19T^{2} \) |
| 23 | \( 1 - 4.71T + 23T^{2} \) |
| 29 | \( 1 - 8.91T + 29T^{2} \) |
| 31 | \( 1 + 2.32T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 7.34T + 41T^{2} \) |
| 43 | \( 1 - 1.47T + 43T^{2} \) |
| 47 | \( 1 + 2.41T + 47T^{2} \) |
| 53 | \( 1 + 2.63T + 53T^{2} \) |
| 59 | \( 1 + 9.21T + 59T^{2} \) |
| 61 | \( 1 + 0.825T + 61T^{2} \) |
| 67 | \( 1 + 5.59T + 67T^{2} \) |
| 71 | \( 1 + 4.69T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 - 9.21T + 89T^{2} \) |
| 97 | \( 1 + 6.86T + 97T^{2} \) |
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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
−7.904169703315243279593324702496, −7.34221458570456968950551101167, −6.14721309614401189718203793614, −5.70164861488776168746406413999, −4.97699489409679449487250838463, −4.73041777061881722946128621128, −3.50620725418908502472059957445, −2.69034538578258957709791395021, −1.43578117433644985203194367687, −0.834807676463820897916219102771,
0.834807676463820897916219102771, 1.43578117433644985203194367687, 2.69034538578258957709791395021, 3.50620725418908502472059957445, 4.73041777061881722946128621128, 4.97699489409679449487250838463, 5.70164861488776168746406413999, 6.14721309614401189718203793614, 7.34221458570456968950551101167, 7.904169703315243279593324702496