| L(s) = 1 | + 3.58·2-s − 2.57·3-s + 4.83·4-s − 5·5-s − 9.20·6-s − 25.2·7-s − 11.3·8-s − 20.3·9-s − 17.9·10-s − 11.7·11-s − 12.4·12-s + 21.3·13-s − 90.5·14-s + 12.8·15-s − 79.3·16-s − 73.0·18-s − 148.·19-s − 24.1·20-s + 64.9·21-s − 42.2·22-s + 7.58·23-s + 29.1·24-s + 25·25-s + 76.5·26-s + 121.·27-s − 122.·28-s + 5.09·29-s + ⋯ |
| L(s) = 1 | + 1.26·2-s − 0.494·3-s + 0.604·4-s − 0.447·5-s − 0.626·6-s − 1.36·7-s − 0.501·8-s − 0.755·9-s − 0.566·10-s − 0.323·11-s − 0.298·12-s + 0.455·13-s − 1.72·14-s + 0.221·15-s − 1.23·16-s − 0.956·18-s − 1.79·19-s − 0.270·20-s + 0.675·21-s − 0.409·22-s + 0.0688·23-s + 0.248·24-s + 0.200·25-s + 0.577·26-s + 0.868·27-s − 0.824·28-s + 0.0326·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9051183769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9051183769\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 3.58T + 8T^{2} \) |
| 3 | \( 1 + 2.57T + 27T^{2} \) |
| 7 | \( 1 + 25.2T + 343T^{2} \) |
| 11 | \( 1 + 11.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.3T + 2.19e3T^{2} \) |
| 19 | \( 1 + 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 7.58T + 1.21e4T^{2} \) |
| 29 | \( 1 - 5.09T + 2.43e4T^{2} \) |
| 31 | \( 1 - 157.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 425.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 381.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 146.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 500.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 234.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 425.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 309.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 191.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 768.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 447.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 313.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 276.T + 9.12e5T^{2} \) |
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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
−8.979910836034809854560149177977, −8.525290129811745562315717741040, −7.14670894227656819799640552476, −6.24972644468804199337040167345, −5.99179188061927498464366995098, −4.93325397823829031224880771974, −4.06590008590434078193968329378, −3.27646961113365497991468832767, −2.48107709806424456970775624935, −0.36826532369273353141008972422,
0.36826532369273353141008972422, 2.48107709806424456970775624935, 3.27646961113365497991468832767, 4.06590008590434078193968329378, 4.93325397823829031224880771974, 5.99179188061927498464366995098, 6.24972644468804199337040167345, 7.14670894227656819799640552476, 8.525290129811745562315717741040, 8.979910836034809854560149177977
