L(s) = 1 | + (0.607 + 1.46i)5-s + (0.258 + 0.965i)9-s + (0.965 + 0.258i)13-s + (0.866 − 0.5i)17-s + (−1.07 + 1.07i)25-s + (1.20 − 1.57i)29-s + (−0.965 − 0.741i)37-s + (−1.96 + 0.258i)41-s + (−1.25 + 0.965i)45-s + (−0.258 + 0.965i)49-s + (0.366 + 0.366i)53-s + (0.965 + 1.25i)61-s + (0.207 + 1.57i)65-s + (−0.758 − 1.83i)73-s + (−0.866 + 0.499i)81-s + ⋯ |
L(s) = 1 | + (0.607 + 1.46i)5-s + (0.258 + 0.965i)9-s + (0.965 + 0.258i)13-s + (0.866 − 0.5i)17-s + (−1.07 + 1.07i)25-s + (1.20 − 1.57i)29-s + (−0.965 − 0.741i)37-s + (−1.96 + 0.258i)41-s + (−1.25 + 0.965i)45-s + (−0.258 + 0.965i)49-s + (0.366 + 0.366i)53-s + (0.965 + 1.25i)61-s + (0.207 + 1.57i)65-s + (−0.758 − 1.83i)73-s + (−0.866 + 0.499i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.553230534\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.553230534\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (-0.965 - 0.258i)T \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
good | 3 | \( 1 + (-0.258 - 0.965i)T^{2} \) |
| 5 | \( 1 + (-0.607 - 1.46i)T + (-0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (0.258 - 0.965i)T^{2} \) |
| 11 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 29 | \( 1 + (-1.20 + 1.57i)T + (-0.258 - 0.965i)T^{2} \) |
| 31 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (0.965 + 0.741i)T + (0.258 + 0.965i)T^{2} \) |
| 41 | \( 1 + (1.96 - 0.258i)T + (0.965 - 0.258i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.366 - 0.366i)T + iT^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.965 - 1.25i)T + (-0.258 + 0.965i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.965 + 0.258i)T^{2} \) |
| 73 | \( 1 + (0.758 + 1.83i)T + (-0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.83 - 0.241i)T + (0.965 + 0.258i)T^{2} \) |
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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
−8.864474508038377510245655282464, −8.044372007175543834330731450415, −7.35073449159369107467738488127, −6.66812849271005722108810855382, −6.01299405275265723196268333217, −5.26232307620952758617218352520, −4.21691717266520532427907270500, −3.23787023778632898741994683677, −2.52632131110962892203888922610, −1.57837384661823409355897206984,
1.04261034801445683776672441990, 1.64335823476480097958879517586, 3.24246784781069945848501611434, 3.89401266857375741713653921937, 5.02189424000186404767356132448, 5.41010876824859071910580181684, 6.36177168966771144164425346337, 6.93631877599294077858031001357, 8.310183646171971938722953556865, 8.508969724381395709298390917951