L(s) = 1 | + 0.810·2-s − 0.0806·3-s − 1.34·4-s + 5-s − 0.0653·6-s − 2.66·7-s − 2.70·8-s − 2.99·9-s + 0.810·10-s + 0.846·11-s + 0.108·12-s − 2.68·13-s − 2.15·14-s − 0.0806·15-s + 0.490·16-s + 3.80·17-s − 2.42·18-s − 1.65·19-s − 1.34·20-s + 0.214·21-s + 0.686·22-s + 3.67·23-s + 0.218·24-s + 25-s − 2.17·26-s + 0.483·27-s + 3.57·28-s + ⋯ |
L(s) = 1 | + 0.573·2-s − 0.0465·3-s − 0.671·4-s + 0.447·5-s − 0.0266·6-s − 1.00·7-s − 0.957·8-s − 0.997·9-s + 0.256·10-s + 0.255·11-s + 0.0312·12-s − 0.745·13-s − 0.576·14-s − 0.0208·15-s + 0.122·16-s + 0.922·17-s − 0.571·18-s − 0.379·19-s − 0.300·20-s + 0.0468·21-s + 0.146·22-s + 0.765·23-s + 0.0445·24-s + 0.200·25-s − 0.427·26-s + 0.0929·27-s + 0.676·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.440902935\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.440902935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 0.810T + 2T^{2} \) |
| 3 | \( 1 + 0.0806T + 3T^{2} \) |
| 7 | \( 1 + 2.66T + 7T^{2} \) |
| 11 | \( 1 - 0.846T + 11T^{2} \) |
| 13 | \( 1 + 2.68T + 13T^{2} \) |
| 17 | \( 1 - 3.80T + 17T^{2} \) |
| 19 | \( 1 + 1.65T + 19T^{2} \) |
| 23 | \( 1 - 3.67T + 23T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 - 8.93T + 31T^{2} \) |
| 37 | \( 1 - 4.44T + 37T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 - 6.96T + 43T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 - 0.743T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 - 6.88T + 67T^{2} \) |
| 71 | \( 1 + 3.28T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 0.745T + 79T^{2} \) |
| 83 | \( 1 - 0.345T + 83T^{2} \) |
| 89 | \( 1 - 3.91T + 89T^{2} \) |
| 97 | \( 1 - 5.92T + 97T^{2} \) |
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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
−9.218412430691534803986654426430, −8.544547516206221374713127322690, −7.60650771308422007994387740151, −6.36876852435798913476707241170, −6.03857053534242528105769549593, −5.10397221815294148968494706283, −4.35585996545991847959126601807, −3.14737162376881430778902712762, −2.72376340677201062428327803079, −0.72199884147993809593394097408,
0.72199884147993809593394097408, 2.72376340677201062428327803079, 3.14737162376881430778902712762, 4.35585996545991847959126601807, 5.10397221815294148968494706283, 6.03857053534242528105769549593, 6.36876852435798913476707241170, 7.60650771308422007994387740151, 8.544547516206221374713127322690, 9.218412430691534803986654426430