L(s) = 1 | + (1.14 − 1.14i)2-s + (0.453 − 0.891i)3-s − 1.61i·4-s + (−0.5 − 1.53i)6-s + (−0.831 − 0.831i)7-s + (−0.707 − 0.707i)8-s + (−0.587 − 0.809i)9-s + (−1.44 − 0.734i)12-s − 1.90·14-s + (−0.437 + 0.437i)17-s + (−1.59 − 0.253i)18-s + (−1.11 + 0.363i)21-s + (−0.951 + 0.309i)24-s + (−0.987 + 0.156i)27-s + (−1.34 + 1.34i)28-s + ⋯ |
L(s) = 1 | + (1.14 − 1.14i)2-s + (0.453 − 0.891i)3-s − 1.61i·4-s + (−0.5 − 1.53i)6-s + (−0.831 − 0.831i)7-s + (−0.707 − 0.707i)8-s + (−0.587 − 0.809i)9-s + (−1.44 − 0.734i)12-s − 1.90·14-s + (−0.437 + 0.437i)17-s + (−1.59 − 0.253i)18-s + (−1.11 + 0.363i)21-s + (−0.951 + 0.309i)24-s + (−0.987 + 0.156i)27-s + (−1.34 + 1.34i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.308445055\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.308445055\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.453 + 0.891i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-1.14 + 1.14i)T - iT^{2} \) |
| 7 | \( 1 + (0.831 + 0.831i)T + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (0.437 - 0.437i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1.34 + 1.34i)T + iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1.14 - 1.14i)T + iT^{2} \) |
| 59 | \( 1 - 1.17T + T^{2} \) |
| 61 | \( 1 - 1.61T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - 1.90iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 0.618iT - T^{2} \) |
| 83 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 89 | \( 1 - 1.17T + T^{2} \) |
| 97 | \( 1 + (-0.831 - 0.831i)T + iT^{2} \) |
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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
−8.435772965644911898294816277516, −7.35410565724013852656512009252, −6.86084015861444980389600055028, −5.97509227735655964248774554838, −5.26699849941063075741416748531, −3.93148000053939637483333417609, −3.76953863564185260987855469048, −2.71347841908458053413900050911, −2.01092967131162696508748563037, −0.859706433913556267121193979715,
2.35722145502626766949872242041, 3.21757767265092281642443479729, 3.83178410998421852210960180882, 4.77644789880642852172413310944, 5.29935116511432646846972375414, 6.03897518826910418097283019702, 6.73396706419759330861348535979, 7.49402217785291230043236833052, 8.460692568687907458274496096683, 8.857177301216428591380640365090