| L(s) = 1 | + 1.45·2-s + 1.71·3-s + 0.114·4-s + 2.59·5-s + 2.48·6-s − 2.00·7-s − 2.74·8-s − 0.0719·9-s + 3.77·10-s − 11-s + 0.196·12-s − 4.45·13-s − 2.92·14-s + 4.43·15-s − 4.21·16-s + 4.54·17-s − 0.104·18-s + 19-s + 0.297·20-s − 3.43·21-s − 1.45·22-s + 7.48·23-s − 4.69·24-s + 1.72·25-s − 6.48·26-s − 5.25·27-s − 0.230·28-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 0.987·3-s + 0.0572·4-s + 1.15·5-s + 1.01·6-s − 0.759·7-s − 0.969·8-s − 0.0239·9-s + 1.19·10-s − 0.301·11-s + 0.0566·12-s − 1.23·13-s − 0.780·14-s + 1.14·15-s − 1.05·16-s + 1.10·17-s − 0.0246·18-s + 0.229·19-s + 0.0664·20-s − 0.750·21-s − 0.310·22-s + 1.56·23-s − 0.957·24-s + 0.344·25-s − 1.27·26-s − 1.01·27-s − 0.0435·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.360535372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.360535372\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.45T + 2T^{2} \) |
| 3 | \( 1 - 1.71T + 3T^{2} \) |
| 5 | \( 1 - 2.59T + 5T^{2} \) |
| 7 | \( 1 + 2.00T + 7T^{2} \) |
| 13 | \( 1 + 4.45T + 13T^{2} \) |
| 17 | \( 1 - 4.54T + 17T^{2} \) |
| 23 | \( 1 - 7.48T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 9.34T + 31T^{2} \) |
| 37 | \( 1 + 6.84T + 37T^{2} \) |
| 41 | \( 1 - 0.644T + 41T^{2} \) |
| 43 | \( 1 + 8.07T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 5.53T + 53T^{2} \) |
| 59 | \( 1 - 9.16T + 59T^{2} \) |
| 61 | \( 1 - 9.45T + 61T^{2} \) |
| 67 | \( 1 - 0.113T + 67T^{2} \) |
| 71 | \( 1 + 9.84T + 71T^{2} \) |
| 73 | \( 1 + 2.38T + 73T^{2} \) |
| 79 | \( 1 + 2.01T + 79T^{2} \) |
| 83 | \( 1 + 2.90T + 83T^{2} \) |
| 89 | \( 1 + 8.82T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−12.81941977052185907269869843554, −11.82973936214951436535960108712, −10.05245212512148806914458653181, −9.537159741269371160089511199744, −8.583668575083022031319086831428, −7.12291854962630392822883661181, −5.84624693367181628228119132762, −5.00755147600689311263117885575, −3.33603189733119050953858467186, −2.56246171539699370489019396293,
2.56246171539699370489019396293, 3.33603189733119050953858467186, 5.00755147600689311263117885575, 5.84624693367181628228119132762, 7.12291854962630392822883661181, 8.583668575083022031319086831428, 9.537159741269371160089511199744, 10.05245212512148806914458653181, 11.82973936214951436535960108712, 12.81941977052185907269869843554
