| L(s) = 1 | − 1.01·2-s − 0.720·3-s − 0.977·4-s − 2.17·5-s + 0.728·6-s + 3.01·8-s − 2.48·9-s + 2.20·10-s + 1.36·11-s + 0.704·12-s + 5.86·13-s + 1.56·15-s − 1.08·16-s + 6.06·17-s + 2.50·18-s − 2.16·19-s + 2.13·20-s − 1.37·22-s + 3.47·23-s − 2.16·24-s − 0.250·25-s − 5.92·26-s + 3.94·27-s + 6.75·29-s − 1.58·30-s − 1.18·31-s − 4.92·32-s + ⋯ |
| L(s) = 1 | − 0.714·2-s − 0.415·3-s − 0.488·4-s − 0.974·5-s + 0.297·6-s + 1.06·8-s − 0.827·9-s + 0.696·10-s + 0.410·11-s + 0.203·12-s + 1.62·13-s + 0.405·15-s − 0.272·16-s + 1.46·17-s + 0.591·18-s − 0.496·19-s + 0.476·20-s − 0.293·22-s + 0.724·23-s − 0.442·24-s − 0.0501·25-s − 1.16·26-s + 0.759·27-s + 1.25·29-s − 0.289·30-s − 0.213·31-s − 0.869·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8416158781\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8416158781\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 1.01T + 2T^{2} \) |
| 3 | \( 1 + 0.720T + 3T^{2} \) |
| 5 | \( 1 + 2.17T + 5T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 - 5.86T + 13T^{2} \) |
| 17 | \( 1 - 6.06T + 17T^{2} \) |
| 19 | \( 1 + 2.16T + 19T^{2} \) |
| 23 | \( 1 - 3.47T + 23T^{2} \) |
| 29 | \( 1 - 6.75T + 29T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 0.769T + 41T^{2} \) |
| 43 | \( 1 - 4.36T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 8.26T + 53T^{2} \) |
| 59 | \( 1 - 5.53T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 8.19T + 67T^{2} \) |
| 71 | \( 1 + 8.12T + 71T^{2} \) |
| 73 | \( 1 - 5.55T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 1.64T + 83T^{2} \) |
| 89 | \( 1 - 4.15T + 89T^{2} \) |
| 97 | \( 1 + 18.3T + 97T^{2} \) |
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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
−8.284977766717075559888545168691, −7.60498865323333887529782367120, −6.70046012135292606659598467056, −5.93288361125739399883971655062, −5.25244426603089707171727038524, −4.28606263748988136092463417395, −3.75103504290166297684480257811, −2.89609470311155891553611713471, −1.27849461582733175371559952407, −0.64796164840445423741582083037,
0.64796164840445423741582083037, 1.27849461582733175371559952407, 2.89609470311155891553611713471, 3.75103504290166297684480257811, 4.28606263748988136092463417395, 5.25244426603089707171727038524, 5.93288361125739399883971655062, 6.70046012135292606659598467056, 7.60498865323333887529782367120, 8.284977766717075559888545168691
