| L(s) = 1 | − 2.89·3-s + 5-s + 7-s + 5.40·9-s + 5.42·11-s + 13-s − 2.89·15-s − 0.135·17-s + 5.37·19-s − 2.89·21-s + 8.82·23-s + 25-s − 6.96·27-s − 2.23·29-s + 9.93·31-s − 15.7·33-s + 35-s + 4.70·37-s − 2.89·39-s − 5.16·41-s − 4.56·43-s + 5.40·45-s − 6.35·47-s + 49-s + 0.393·51-s + 3.97·53-s + 5.42·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s + 0.447·5-s + 0.377·7-s + 1.80·9-s + 1.63·11-s + 0.277·13-s − 0.748·15-s − 0.0329·17-s + 1.23·19-s − 0.632·21-s + 1.84·23-s + 0.200·25-s − 1.33·27-s − 0.415·29-s + 1.78·31-s − 2.73·33-s + 0.169·35-s + 0.772·37-s − 0.464·39-s − 0.806·41-s − 0.696·43-s + 0.805·45-s − 0.927·47-s + 0.142·49-s + 0.0551·51-s + 0.546·53-s + 0.731·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.581348738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.581348738\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.89T + 3T^{2} \) |
| 11 | \( 1 - 5.42T + 11T^{2} \) |
| 17 | \( 1 + 0.135T + 17T^{2} \) |
| 19 | \( 1 - 5.37T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 + 2.23T + 29T^{2} \) |
| 31 | \( 1 - 9.93T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 + 5.16T + 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 + 6.35T + 47T^{2} \) |
| 53 | \( 1 - 3.97T + 53T^{2} \) |
| 59 | \( 1 - 4.97T + 59T^{2} \) |
| 61 | \( 1 + 6.24T + 61T^{2} \) |
| 67 | \( 1 + 8.50T + 67T^{2} \) |
| 71 | \( 1 - 2.10T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−8.663587968172113107018260454758, −7.56041085933736986318062976761, −6.60904973818215937425565447133, −6.50168252249531647822308451606, −5.48290558440772875874264910980, −4.95634364870212536593792235310, −4.19309536486409983689204210974, −3.08131794349135441747663102187, −1.45545064135512161157278045758, −0.941063827821947588673892513371,
0.941063827821947588673892513371, 1.45545064135512161157278045758, 3.08131794349135441747663102187, 4.19309536486409983689204210974, 4.95634364870212536593792235310, 5.48290558440772875874264910980, 6.50168252249531647822308451606, 6.60904973818215937425565447133, 7.56041085933736986318062976761, 8.663587968172113107018260454758
