L(s) = 1 | − 2-s + 3-s + 4-s − 3.67·5-s − 6-s − 3.11·7-s − 8-s + 9-s + 3.67·10-s + 2.60·11-s + 12-s + 13-s + 3.11·14-s − 3.67·15-s + 16-s − 3.87·17-s − 18-s − 8.68·19-s − 3.67·20-s − 3.11·21-s − 2.60·22-s + 1.84·23-s − 24-s + 8.50·25-s − 26-s + 27-s − 3.11·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.64·5-s − 0.408·6-s − 1.17·7-s − 0.353·8-s + 0.333·9-s + 1.16·10-s + 0.786·11-s + 0.288·12-s + 0.277·13-s + 0.832·14-s − 0.948·15-s + 0.250·16-s − 0.938·17-s − 0.235·18-s − 1.99·19-s − 0.821·20-s − 0.679·21-s − 0.556·22-s + 0.383·23-s − 0.204·24-s + 1.70·25-s − 0.196·26-s + 0.192·27-s − 0.588·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4187600890\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4187600890\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 3.67T + 5T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 17 | \( 1 + 3.87T + 17T^{2} \) |
| 19 | \( 1 + 8.68T + 19T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 29 | \( 1 - 2.03T + 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 - 0.985T + 37T^{2} \) |
| 41 | \( 1 + 6.82T + 41T^{2} \) |
| 43 | \( 1 - 3.73T + 43T^{2} \) |
| 47 | \( 1 + 0.0401T + 47T^{2} \) |
| 53 | \( 1 + 3.86T + 53T^{2} \) |
| 59 | \( 1 + 12.0T + 59T^{2} \) |
| 61 | \( 1 + 9.47T + 61T^{2} \) |
| 67 | \( 1 + 5.87T + 67T^{2} \) |
| 71 | \( 1 - 7.23T + 71T^{2} \) |
| 73 | \( 1 - 2.66T + 73T^{2} \) |
| 79 | \( 1 + 0.518T + 79T^{2} \) |
| 83 | \( 1 - 3.79T + 83T^{2} \) |
| 89 | \( 1 - 1.83T + 89T^{2} \) |
| 97 | \( 1 + 2.45T + 97T^{2} \) |
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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
−7.87812723304935628252624475479, −7.27614604288166979102198581415, −6.53261117316157913272565824925, −6.29801050626113133422394195675, −4.72983905189524137939004504213, −4.02939535669339915624503366770, −3.52283735986832368666103700179, −2.77039432448796744722966932295, −1.70457262574009312378176944755, −0.33553933816055386407542027941,
0.33553933816055386407542027941, 1.70457262574009312378176944755, 2.77039432448796744722966932295, 3.52283735986832368666103700179, 4.02939535669339915624503366770, 4.72983905189524137939004504213, 6.29801050626113133422394195675, 6.53261117316157913272565824925, 7.27614604288166979102198581415, 7.87812723304935628252624475479