| L(s) = 1 | − 1.82·2-s + 2.83·3-s + 1.31·4-s + 5-s − 5.15·6-s + 3.11·7-s + 1.24·8-s + 5.01·9-s − 1.82·10-s − 3.80·11-s + 3.71·12-s − 4.66·13-s − 5.67·14-s + 2.83·15-s − 4.90·16-s − 9.13·18-s + 3.26·19-s + 1.31·20-s + 8.82·21-s + 6.91·22-s + 3.45·23-s + 3.53·24-s + 25-s + 8.48·26-s + 5.71·27-s + 4.09·28-s + 8.09·29-s + ⋯ |
| L(s) = 1 | − 1.28·2-s + 1.63·3-s + 0.656·4-s + 0.447·5-s − 2.10·6-s + 1.17·7-s + 0.441·8-s + 1.67·9-s − 0.575·10-s − 1.14·11-s + 1.07·12-s − 1.29·13-s − 1.51·14-s + 0.731·15-s − 1.22·16-s − 2.15·18-s + 0.749·19-s + 0.293·20-s + 1.92·21-s + 1.47·22-s + 0.720·23-s + 0.722·24-s + 0.200·25-s + 1.66·26-s + 1.09·27-s + 0.773·28-s + 1.50·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801982014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801982014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 3 | \( 1 - 2.83T + 3T^{2} \) |
| 7 | \( 1 - 3.11T + 7T^{2} \) |
| 11 | \( 1 + 3.80T + 11T^{2} \) |
| 13 | \( 1 + 4.66T + 13T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 - 3.45T + 23T^{2} \) |
| 29 | \( 1 - 8.09T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 - 0.684T + 37T^{2} \) |
| 41 | \( 1 - 0.750T + 41T^{2} \) |
| 43 | \( 1 - 0.425T + 43T^{2} \) |
| 47 | \( 1 + 1.28T + 47T^{2} \) |
| 53 | \( 1 + 3.38T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 4.44T + 83T^{2} \) |
| 89 | \( 1 + 1.20T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−9.532102751719962543554139599553, −8.539397567201359927003822983149, −8.145498485462681218022110827613, −7.63420948160484542970414806563, −6.86154963617970139831252267644, −5.08025619097273792288731147997, −4.54893938253515031526606164447, −2.84404425695411457747647576371, −2.31611924987053397415744989037, −1.17850270895077677836140893519,
1.17850270895077677836140893519, 2.31611924987053397415744989037, 2.84404425695411457747647576371, 4.54893938253515031526606164447, 5.08025619097273792288731147997, 6.86154963617970139831252267644, 7.63420948160484542970414806563, 8.145498485462681218022110827613, 8.539397567201359927003822983149, 9.532102751719962543554139599553
