L(s) = 1 | − 2-s + 3-s + 4-s + 3.85·5-s − 6-s + 3.88·7-s − 8-s + 9-s − 3.85·10-s − 5.32·11-s + 12-s − 13-s − 3.88·14-s + 3.85·15-s + 16-s + 0.277·17-s − 18-s + 3.65·19-s + 3.85·20-s + 3.88·21-s + 5.32·22-s + 0.824·23-s − 24-s + 9.88·25-s + 26-s + 27-s + 3.88·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.72·5-s − 0.408·6-s + 1.46·7-s − 0.353·8-s + 0.333·9-s − 1.22·10-s − 1.60·11-s + 0.288·12-s − 0.277·13-s − 1.03·14-s + 0.996·15-s + 0.250·16-s + 0.0672·17-s − 0.235·18-s + 0.839·19-s + 0.862·20-s + 0.846·21-s + 1.13·22-s + 0.171·23-s − 0.204·24-s + 1.97·25-s + 0.196·26-s + 0.192·27-s + 0.733·28-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
\(L(1)\) |
\(\approx\) |
\(3.205506946\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.205506946\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.85T + 5T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 17 | \( 1 - 0.277T + 17T^{2} \) |
| 19 | \( 1 - 3.65T + 19T^{2} \) |
| 23 | \( 1 - 0.824T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 4.97T + 31T^{2} \) |
| 37 | \( 1 - 2.46T + 37T^{2} \) |
| 41 | \( 1 - 5.80T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 - 4.44T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 2.38T + 61T^{2} \) |
| 67 | \( 1 + 7.40T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 1.14T + 79T^{2} \) |
| 83 | \( 1 - 6.90T + 83T^{2} \) |
| 89 | \( 1 - 8.07T + 89T^{2} \) |
| 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
−8.005722787931549692328970402361, −7.35722556889490968489473171243, −6.57604025919704633803028417148, −5.58543233645339145353914412207, −5.23511673172195673893871307644, −4.53049773788525640087183808576, −2.99348088182784422880466715165, −2.46927746563317339293535643619, −1.81151599460819881026826891072, −1.01628015381954244979385961010,
1.01628015381954244979385961010, 1.81151599460819881026826891072, 2.46927746563317339293535643619, 2.99348088182784422880466715165, 4.53049773788525640087183808576, 5.23511673172195673893871307644, 5.58543233645339145353914412207, 6.57604025919704633803028417148, 7.35722556889490968489473171243, 8.005722787931549692328970402361