| L(s) = 1 | − 1.58·2-s − 3-s + 0.497·4-s − 0.196·5-s + 1.58·6-s − 1.13·7-s + 2.37·8-s + 9-s + 0.311·10-s − 0.497·12-s − 2.42·13-s + 1.80·14-s + 0.196·15-s − 4.74·16-s − 6.24·17-s − 1.58·18-s − 19-s − 0.0979·20-s + 1.13·21-s − 7.08·23-s − 2.37·24-s − 4.96·25-s + 3.83·26-s − 27-s − 0.566·28-s − 7.16·29-s − 0.311·30-s + ⋯ |
| L(s) = 1 | − 1.11·2-s − 0.577·3-s + 0.248·4-s − 0.0880·5-s + 0.645·6-s − 0.430·7-s + 0.839·8-s + 0.333·9-s + 0.0984·10-s − 0.143·12-s − 0.672·13-s + 0.481·14-s + 0.0508·15-s − 1.18·16-s − 1.51·17-s − 0.372·18-s − 0.229·19-s − 0.0219·20-s + 0.248·21-s − 1.47·23-s − 0.484·24-s − 0.992·25-s + 0.751·26-s − 0.192·27-s − 0.107·28-s − 1.33·29-s − 0.0568·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6897 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05146847054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05146847054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 1.58T + 2T^{2} \) |
| 5 | \( 1 + 0.196T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 13 | \( 1 + 2.42T + 13T^{2} \) |
| 17 | \( 1 + 6.24T + 17T^{2} \) |
| 23 | \( 1 + 7.08T + 23T^{2} \) |
| 29 | \( 1 + 7.16T + 29T^{2} \) |
| 31 | \( 1 + 2.66T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 41 | \( 1 - 1.28T + 41T^{2} \) |
| 43 | \( 1 + 6.56T + 43T^{2} \) |
| 47 | \( 1 - 5.57T + 47T^{2} \) |
| 53 | \( 1 + 2.30T + 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 - 1.53T + 67T^{2} \) |
| 71 | \( 1 - 7.71T + 71T^{2} \) |
| 73 | \( 1 - 8.13T + 73T^{2} \) |
| 79 | \( 1 - 2.13T + 79T^{2} \) |
| 83 | \( 1 + 9.23T + 83T^{2} \) |
| 89 | \( 1 + 7.89T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.80999476588707549337540726809, −7.55551854073796179981666519959, −6.61252398949079767474804658601, −6.09725223849837241358493884457, −5.12117491423832131596677791092, −4.37301141045371904208358181231, −3.73120850815251272185439089944, −2.30345124477504484215884239628, −1.67264642020769872529646118833, −0.13821455734791869098032231666,
0.13821455734791869098032231666, 1.67264642020769872529646118833, 2.30345124477504484215884239628, 3.73120850815251272185439089944, 4.37301141045371904208358181231, 5.12117491423832131596677791092, 6.09725223849837241358493884457, 6.61252398949079767474804658601, 7.55551854073796179981666519959, 7.80999476588707549337540726809
