L(s) = 1 | + (−0.809 − 0.587i)4-s + (0.831 − 1.14i)7-s + (−1.34 − 0.437i)13-s + (0.309 + 0.951i)16-s + (−0.831 − 1.14i)19-s + (0.809 − 0.587i)25-s + (−1.34 + 0.437i)28-s − 1.41i·43-s + (−0.309 − 0.951i)49-s + (0.831 + 1.14i)52-s + (1.34 − 0.437i)61-s + (0.309 − 0.951i)64-s + (−0.831 + 1.14i)73-s + 1.41i·76-s + (1.34 + 0.437i)79-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)4-s + (0.831 − 1.14i)7-s + (−1.34 − 0.437i)13-s + (0.309 + 0.951i)16-s + (−0.831 − 1.14i)19-s + (0.809 − 0.587i)25-s + (−1.34 + 0.437i)28-s − 1.41i·43-s + (−0.309 − 0.951i)49-s + (0.831 + 1.14i)52-s + (1.34 − 0.437i)61-s + (0.309 − 0.951i)64-s + (−0.831 + 1.14i)73-s + 1.41i·76-s + (1.34 + 0.437i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0938 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0938 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7977144954\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7977144954\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + 1.41iT - T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.34 + 0.437i)T + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.00287834441950506099861226647, −9.030564040896637976408327841118, −8.265057042609235963932222246621, −7.37456768357368292289237611418, −6.58894840832320962476003324715, −5.22346012892849327190364256642, −4.74484380664329132083583427079, −3.90734582682606064973369848354, −2.31917304404941302629406842304, −0.76198671179119282246770457432,
1.94848483513618685276119927009, 3.05361966177090638218079065431, 4.36092048291769080420979355836, 4.99048785404188417271629906794, 5.87022071246022123282800388836, 7.14052110300060959759244997026, 8.009256356972757416304581333066, 8.607123413207573461843242586894, 9.335315909755807050048303636413, 10.08378027266777955122824549369