| L(s) = 1 | − 2.37·2-s + 3-s + 3.66·4-s − 2.66·5-s − 2.37·6-s + 7-s − 3.95·8-s + 9-s + 6.33·10-s + 1.57·11-s + 3.66·12-s + 13-s − 2.37·14-s − 2.66·15-s + 2.08·16-s + 4.75·17-s − 2.37·18-s − 2.23·19-s − 9.74·20-s + 21-s − 3.74·22-s + 5.84·23-s − 3.95·24-s + 2.08·25-s − 2.37·26-s + 27-s + 3.66·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.577·3-s + 1.83·4-s − 1.19·5-s − 0.971·6-s + 0.377·7-s − 1.39·8-s + 0.333·9-s + 2.00·10-s + 0.475·11-s + 1.05·12-s + 0.277·13-s − 0.635·14-s − 0.687·15-s + 0.521·16-s + 1.15·17-s − 0.560·18-s − 0.513·19-s − 2.17·20-s + 0.218·21-s − 0.799·22-s + 1.21·23-s − 0.807·24-s + 0.417·25-s − 0.466·26-s + 0.192·27-s + 0.692·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6459771568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6459771568\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 11 | \( 1 - 1.57T + 11T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 19 | \( 1 + 2.23T + 19T^{2} \) |
| 23 | \( 1 - 5.84T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 2.25T + 41T^{2} \) |
| 43 | \( 1 - 0.913T + 43T^{2} \) |
| 47 | \( 1 + 2.09T + 47T^{2} \) |
| 53 | \( 1 - 1.08T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 + 7.51T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 7.42T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−11.57545675862386380276952236215, −10.81245258030240260246219271925, −9.813603607650586946200993676002, −8.880283484154178215276845930752, −8.090083866786603322353590624015, −7.59725820643528041546968534191, −6.48750008931052689296052616965, −4.43190800443986335732644538989, −2.96959373929293272112849619855, −1.13463035130736646229008723762,
1.13463035130736646229008723762, 2.96959373929293272112849619855, 4.43190800443986335732644538989, 6.48750008931052689296052616965, 7.59725820643528041546968534191, 8.090083866786603322353590624015, 8.880283484154178215276845930752, 9.813603607650586946200993676002, 10.81245258030240260246219271925, 11.57545675862386380276952236215
