| L(s) = 1 | + 1.30i·2-s + 0.302·4-s + 4.60i·7-s + 3i·8-s − 2.60·11-s + 0.605i·13-s − 6·14-s − 3.30·16-s − 5.60i·17-s + 3.60·19-s − 3.39i·22-s + 3i·23-s − 0.788·26-s + 1.39i·28-s − 8.60·29-s + ⋯ |
| L(s) = 1 | + 0.921i·2-s + 0.151·4-s + 1.74i·7-s + 1.06i·8-s − 0.785·11-s + 0.167i·13-s − 1.60·14-s − 0.825·16-s − 1.35i·17-s + 0.827·19-s − 0.723i·22-s + 0.625i·23-s − 0.154·26-s + 0.263i·28-s − 1.59·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.338748 + 1.43496i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.338748 + 1.43496i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.30iT - 2T^{2} \) |
| 7 | \( 1 - 4.60iT - 7T^{2} \) |
| 11 | \( 1 + 2.60T + 11T^{2} \) |
| 13 | \( 1 - 0.605iT - 13T^{2} \) |
| 17 | \( 1 + 5.60iT - 17T^{2} \) |
| 19 | \( 1 - 3.60T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 + 8.60T + 29T^{2} \) |
| 31 | \( 1 - 1.60T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 2.60T + 41T^{2} \) |
| 43 | \( 1 - 6.60iT - 43T^{2} \) |
| 47 | \( 1 - 5.21iT - 47T^{2} \) |
| 53 | \( 1 - 5.60iT - 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 + 15.2iT - 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 5.39iT - 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 + 3iT - 83T^{2} \) |
| 89 | \( 1 + 7.81T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−11.14504375364893509531644698762, −9.632527891606047458637918206007, −9.091116827626980143934908033710, −8.043414473503956553647949942968, −7.44123834041897679903063706406, −6.35317451912469313849385528827, −5.47532730397400331580432098109, −5.05060326899302559607509480843, −3.03372936300460914596785233955, −2.16857170541265553963773382891,
0.76416499516960759110806516794, 2.10573064634070077603886748970, 3.52642671663629905397195125261, 4.08468834539344009947217866162, 5.50814986835270701986932434841, 6.80501896824354508123223281424, 7.41421904496491870950532291791, 8.358503531972421996858144707183, 9.804615774495050602800096886810, 10.27292103370449098171996126427
