L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.258 − 0.965i)5-s + (0.707 − 0.707i)8-s − i·10-s + (−0.366 + 1.36i)13-s + (0.500 − 0.866i)16-s + (−0.258 − 0.965i)20-s + (−0.866 − 0.499i)25-s + 1.41i·26-s + (0.707 − 1.22i)29-s + (0.258 − 0.965i)32-s + (−1 + i)37-s + (−0.5 − 0.866i)40-s + (−1.22 + 0.707i)41-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (0.258 − 0.965i)5-s + (0.707 − 0.707i)8-s − i·10-s + (−0.366 + 1.36i)13-s + (0.500 − 0.866i)16-s + (−0.258 − 0.965i)20-s + (−0.866 − 0.499i)25-s + 1.41i·26-s + (0.707 − 1.22i)29-s + (0.258 − 0.965i)32-s + (−1 + i)37-s + (−0.5 − 0.866i)40-s + (−1.22 + 0.707i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.086399401\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.086399401\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.258 + 0.965i)T \) |
good | 7 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - 1.41T + T^{2} \) |
| 97 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
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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
−9.656731080358730846113710143698, −8.717378325915981991698902459332, −7.82988219685914492322175405905, −6.74214661012656216437784750582, −6.17593097403503550775060776019, −5.05778694932925308541634052654, −4.60712475163060111899372708999, −3.67997943357391030786871549092, −2.38256177643502493159938384063, −1.43293737938939737523629963426,
1.98351477667380512182141178703, 3.05051686259952760547042442518, 3.59355093683939764285563496747, 4.95069704406478965128708288720, 5.55911920194567413784618488031, 6.47215399947650069250528863775, 7.13951472902241617781450168332, 7.83651076139642707586995853827, 8.743853717371837974354219215498, 10.04128762777582076569372110649