| L(s) = 1 | + 1.13·3-s − 1.30·5-s + 1.71·7-s − 1.70·9-s + 5.30·11-s + 2.18·13-s − 1.48·15-s + 0.392·17-s − 0.498·19-s + 1.95·21-s − 8.33·23-s − 3.29·25-s − 5.35·27-s − 4.50·29-s + 2.96·31-s + 6.04·33-s − 2.23·35-s + 3.06·37-s + 2.49·39-s − 10.0·41-s − 9.93·43-s + 2.22·45-s − 0.714·47-s − 4.06·49-s + 0.447·51-s − 10.3·53-s − 6.93·55-s + ⋯ |
| L(s) = 1 | + 0.657·3-s − 0.584·5-s + 0.647·7-s − 0.567·9-s + 1.60·11-s + 0.606·13-s − 0.384·15-s + 0.0952·17-s − 0.114·19-s + 0.426·21-s − 1.73·23-s − 0.658·25-s − 1.03·27-s − 0.837·29-s + 0.532·31-s + 1.05·33-s − 0.378·35-s + 0.504·37-s + 0.399·39-s − 1.57·41-s − 1.51·43-s + 0.331·45-s − 0.104·47-s − 0.580·49-s + 0.0626·51-s − 1.42·53-s − 0.934·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 547 | \( 1 - T \) |
| good | 3 | \( 1 - 1.13T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 - 5.30T + 11T^{2} \) |
| 13 | \( 1 - 2.18T + 13T^{2} \) |
| 17 | \( 1 - 0.392T + 17T^{2} \) |
| 19 | \( 1 + 0.498T + 19T^{2} \) |
| 23 | \( 1 + 8.33T + 23T^{2} \) |
| 29 | \( 1 + 4.50T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 9.93T + 43T^{2} \) |
| 47 | \( 1 + 0.714T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 + 0.889T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 - 5.35T + 73T^{2} \) |
| 79 | \( 1 - 7.79T + 79T^{2} \) |
| 83 | \( 1 - 17.9T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−7.77053727432857773881845099121, −6.62305147826892817145921930758, −6.22082163137101775850777002782, −5.32960222223303040708364623035, −4.38785420124423598704092746575, −3.74155379306734764654655993141, −3.33184687631926579556446718862, −2.05606028094838902882825806809, −1.47583695162180142329756411178, 0,
1.47583695162180142329756411178, 2.05606028094838902882825806809, 3.33184687631926579556446718862, 3.74155379306734764654655993141, 4.38785420124423598704092746575, 5.32960222223303040708364623035, 6.22082163137101775850777002782, 6.62305147826892817145921930758, 7.77053727432857773881845099121
