L(s) = 1 | + 2-s + 0.0970·3-s + 4-s + 5-s + 0.0970·6-s − 7-s + 8-s − 2.99·9-s + 10-s + 0.0970·12-s − 4.32·13-s − 14-s + 0.0970·15-s + 16-s + 2.12·17-s − 2.99·18-s + 5.51·19-s + 20-s − 0.0970·21-s + 0.922·23-s + 0.0970·24-s + 25-s − 4.32·26-s − 0.581·27-s − 28-s − 8.02·29-s + 0.0970·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.0560·3-s + 0.5·4-s + 0.447·5-s + 0.0396·6-s − 0.377·7-s + 0.353·8-s − 0.996·9-s + 0.316·10-s + 0.0280·12-s − 1.19·13-s − 0.267·14-s + 0.0250·15-s + 0.250·16-s + 0.514·17-s − 0.704·18-s + 1.26·19-s + 0.223·20-s − 0.0211·21-s + 0.192·23-s + 0.0198·24-s + 0.200·25-s − 0.848·26-s − 0.111·27-s − 0.188·28-s − 1.49·29-s + 0.0177·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 0.0970T + 3T^{2} \) |
| 13 | \( 1 + 4.32T + 13T^{2} \) |
| 17 | \( 1 - 2.12T + 17T^{2} \) |
| 19 | \( 1 - 5.51T + 19T^{2} \) |
| 23 | \( 1 - 0.922T + 23T^{2} \) |
| 29 | \( 1 + 8.02T + 29T^{2} \) |
| 31 | \( 1 - 1.21T + 31T^{2} \) |
| 37 | \( 1 + 5.01T + 37T^{2} \) |
| 41 | \( 1 + 1.92T + 41T^{2} \) |
| 43 | \( 1 - 3.75T + 43T^{2} \) |
| 47 | \( 1 + 0.865T + 47T^{2} \) |
| 53 | \( 1 + 0.729T + 53T^{2} \) |
| 59 | \( 1 + 0.178T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 3.55T + 67T^{2} \) |
| 71 | \( 1 - 5.96T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 - 13.1T + 79T^{2} \) |
| 83 | \( 1 - 2.63T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 - 0.0855T + 97T^{2} \) |
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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
−7.40909790216239924201763321882, −6.68686563161900584302091439475, −5.83039617983020942841145089784, −5.41826035597849132970673286331, −4.83064159543155140457918988986, −3.75705404804887472767793536280, −3.04976962692796330837463404373, −2.50016783508527253878429473649, −1.45306128597451646460832657660, 0,
1.45306128597451646460832657660, 2.50016783508527253878429473649, 3.04976962692796330837463404373, 3.75705404804887472767793536280, 4.83064159543155140457918988986, 5.41826035597849132970673286331, 5.83039617983020942841145089784, 6.68686563161900584302091439475, 7.40909790216239924201763321882