| L(s) = 1 | − 1.41i·2-s + (1.91 − 2.31i)3-s − 2.00·4-s + 3.94i·5-s + (−3.26 − 2.70i)6-s + 4.74·7-s + 2.82i·8-s + (−1.68 − 8.84i)9-s + 5.58·10-s − 15.6i·11-s + (−3.82 + 4.62i)12-s − 12.6·13-s − 6.71i·14-s + (9.13 + 7.55i)15-s + 4.00·16-s − 12.7i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (0.637 − 0.770i)3-s − 0.500·4-s + 0.789i·5-s + (−0.544 − 0.450i)6-s + 0.678·7-s + 0.353i·8-s + (−0.187 − 0.982i)9-s + 0.558·10-s − 1.41i·11-s + (−0.318 + 0.385i)12-s − 0.976·13-s − 0.479i·14-s + (0.608 + 0.503i)15-s + 0.250·16-s − 0.749i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.802243 - 1.70454i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.802243 - 1.70454i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-1.91 + 2.31i)T \) |
| 59 | \( 1 - 7.68iT \) |
| good | 5 | \( 1 - 3.94iT - 25T^{2} \) |
| 7 | \( 1 - 4.74T + 49T^{2} \) |
| 11 | \( 1 + 15.6iT - 121T^{2} \) |
| 13 | \( 1 + 12.6T + 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 17.4T + 361T^{2} \) |
| 23 | \( 1 + 43.6iT - 529T^{2} \) |
| 29 | \( 1 + 5.66iT - 841T^{2} \) |
| 31 | \( 1 - 32.2T + 961T^{2} \) |
| 37 | \( 1 + 53.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 32.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 22.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 58.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 78.1iT - 2.80e3T^{2} \) |
| 61 | \( 1 - 66.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 21.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 25.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 4.59T + 5.32e3T^{2} \) |
| 79 | \( 1 - 82.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 46.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 176. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 112.T + 9.40e3T^{2} \) |
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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
−11.06612212187884318085076685004, −10.11153162533453480046040949554, −8.986629320571813752187761577323, −8.187080193074820863459347791942, −7.26061373067951460829150672400, −6.18905557955148174034896879060, −4.75402112333205728854381344800, −3.16085054648858322216178387896, −2.52841753450553893771447557156, −0.838366398687177123178318526583,
1.85299522565182925957523499541, 3.71298847384912094516007859491, 4.93278360115375959054402150768, 5.20740379879510095881274576883, 7.13716407102998720387551082114, 7.86373652762330152969203462020, 8.770221642194060260423936041942, 9.625802753150137362416501247465, 10.23174456510600751991285006949, 11.67403788790038423382871624118
