L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (−1.53 − 0.5i)13-s + (0.309 − 0.951i)16-s + (−0.363 + 0.5i)17-s + 0.999i·18-s − 1.61·26-s + (0.5 − 0.363i)29-s − i·32-s + (−0.190 + 0.587i)34-s + (0.309 + 0.951i)36-s + (−0.587 − 0.190i)37-s + (−0.5 + 1.53i)41-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.587 − 0.809i)8-s + (−0.309 + 0.951i)9-s + (−1.53 − 0.5i)13-s + (0.309 − 0.951i)16-s + (−0.363 + 0.5i)17-s + 0.999i·18-s − 1.61·26-s + (0.5 − 0.363i)29-s − i·32-s + (−0.190 + 0.587i)34-s + (0.309 + 0.951i)36-s + (−0.587 − 0.190i)37-s + (−0.5 + 1.53i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.416232146\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.416232146\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.587 + 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.53 + 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−11.14746061466643168133306605352, −10.39453964008250323297925550671, −9.634848208909771138410520565847, −8.179454630665018872960167910744, −7.35177058835750059944911146903, −6.27603970654745280618525656264, −5.19068489711970881598786626096, −4.54653968938110739632391960999, −3.07971961357864023153702420743, −2.06982402263305816309196240302,
2.31310882973773164640443285196, 3.47466017274630736535937894105, 4.62313276235019003641204582155, 5.49445452418867130326668713630, 6.69862703886274849780181601487, 7.19343088552315732137679538871, 8.437819512607701971991879243673, 9.413112162294430392151272609651, 10.44778531232673743064491779049, 11.64747676121945895011989287284