| L(s) = 1 | − 1.09·2-s − 1.39·3-s − 0.807·4-s − 1.62·5-s + 1.52·6-s + 3.06·8-s − 1.05·9-s + 1.77·10-s + 1.29·11-s + 1.12·12-s + 2.26·15-s − 1.73·16-s + 5.41·17-s + 1.14·18-s − 1.51·19-s + 1.31·20-s − 1.41·22-s − 2.64·23-s − 4.27·24-s − 2.35·25-s + 5.65·27-s + 5.81·29-s − 2.47·30-s + 7.28·31-s − 4.23·32-s − 1.80·33-s − 5.90·34-s + ⋯ |
| L(s) = 1 | − 0.772·2-s − 0.805·3-s − 0.403·4-s − 0.727·5-s + 0.622·6-s + 1.08·8-s − 0.350·9-s + 0.561·10-s + 0.390·11-s + 0.325·12-s + 0.586·15-s − 0.433·16-s + 1.31·17-s + 0.270·18-s − 0.346·19-s + 0.293·20-s − 0.301·22-s − 0.550·23-s − 0.873·24-s − 0.470·25-s + 1.08·27-s + 1.08·29-s − 0.452·30-s + 1.30·31-s − 0.749·32-s − 0.314·33-s − 1.01·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6042068426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6042068426\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.09T + 2T^{2} \) |
| 3 | \( 1 + 1.39T + 3T^{2} \) |
| 5 | \( 1 + 1.62T + 5T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 + 1.51T + 19T^{2} \) |
| 23 | \( 1 + 2.64T + 23T^{2} \) |
| 29 | \( 1 - 5.81T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 - 6.95T + 37T^{2} \) |
| 41 | \( 1 + 8.09T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 7.17T + 47T^{2} \) |
| 53 | \( 1 - 4.66T + 53T^{2} \) |
| 59 | \( 1 - 0.773T + 59T^{2} \) |
| 61 | \( 1 + 8.74T + 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 + 11.9T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 8.42T + 83T^{2} \) |
| 89 | \( 1 + 1.66T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.72792911789640590073492791498, −7.47727290614848447334733171690, −6.30739567541096766403262053037, −5.89762849235619401581026237221, −4.93689194756712455859006501874, −4.37935660453765164003047917938, −3.63208102599414868160534190036, −2.60582284440693825623472518388, −1.22195767521668794103994724812, −0.53227070040051855052438261630,
0.53227070040051855052438261630, 1.22195767521668794103994724812, 2.60582284440693825623472518388, 3.63208102599414868160534190036, 4.37935660453765164003047917938, 4.93689194756712455859006501874, 5.89762849235619401581026237221, 6.30739567541096766403262053037, 7.47727290614848447334733171690, 7.72792911789640590073492791498
