L(s) = 1 | + (0.5 + 0.363i)2-s + (−0.190 − 0.587i)4-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.190 − 0.587i)14-s + (−0.190 − 0.587i)18-s + 0.618·23-s + (0.309 − 0.951i)25-s + (−0.5 + 0.363i)28-s + (0.5 + 1.53i)29-s − 32-s + (−0.190 + 0.587i)36-s + (0.190 + 0.587i)37-s + 1.61·43-s + ⋯ |
L(s) = 1 | + (0.5 + 0.363i)2-s + (−0.190 − 0.587i)4-s + (−0.309 − 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (0.190 − 0.587i)14-s + (−0.190 − 0.587i)18-s + 0.618·23-s + (0.309 − 0.951i)25-s + (−0.5 + 0.363i)28-s + (0.5 + 1.53i)29-s − 32-s + (−0.190 + 0.587i)36-s + (0.190 + 0.587i)37-s + 1.61·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.081540667\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.081540667\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 - 0.618T + T^{2} \) |
| 29 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - 1.61T + T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−10.40723633397701078585086442746, −9.390927655691829200951231578610, −8.705980347195707263985304340992, −7.43493849317758554617055918416, −6.64759637896563446570710197689, −5.95949081239452300372056133530, −4.93401139860206675736752099072, −4.05280921891923670979454313241, −2.98615206407622841477444937199, −0.995642187005966018838618935049,
2.31098301842406691402490740664, 2.97747813421447226816711984537, 4.14831058420496007249022438472, 5.23815177666503801746456736982, 5.86614697832475756468757855419, 7.19066474531819586778023625105, 8.153777731174796036854077056752, 8.772340516140955432492537708856, 9.574669557968612341704915386719, 10.83191713871007816334271543006