| L(s) = 1 | + (−1.61 + 0.618i)3-s − 5-s + (0.381 + 2.61i)7-s + (2.23 − 2.00i)9-s + 0.763i·11-s − 1.23i·13-s + (1.61 − 0.618i)15-s + 4.47·17-s + 7.23i·19-s + (−2.23 − 4i)21-s − 7.70i·23-s + 25-s + (−2.38 + 4.61i)27-s + 4i·29-s − 3.23i·31-s + ⋯ |
| L(s) = 1 | + (−0.934 + 0.356i)3-s − 0.447·5-s + (0.144 + 0.989i)7-s + (0.745 − 0.666i)9-s + 0.230i·11-s − 0.342i·13-s + (0.417 − 0.159i)15-s + 1.08·17-s + 1.66i·19-s + (−0.487 − 0.872i)21-s − 1.60i·23-s + 0.200·25-s + (−0.458 + 0.888i)27-s + 0.742i·29-s − 0.581i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8897992661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8897992661\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.61 - 0.618i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-0.381 - 2.61i)T \) |
| good | 11 | \( 1 - 0.763iT - 11T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 - 7.23iT - 19T^{2} \) |
| 23 | \( 1 + 7.70iT - 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + 3.23iT - 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 53 | \( 1 - 9.23iT - 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 4.94iT - 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 - 6.76iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 7.23T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 14.1iT - 97T^{2} \) |
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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
−9.772749743094194075304034127343, −8.837236527526919782465460527493, −8.028839224247582396863212747368, −7.23384364444081227500280248381, −6.04683960090980914721810476968, −5.72408308608106263254591548046, −4.71536588716408887625747386516, −3.89352038671615396973922179673, −2.74515150645285011540653528066, −1.23599390436585876707035182937,
0.44804905963055870649673793483, 1.53147725854061143074148867374, 3.15419947570887438882220295264, 4.21701139412991818773128086002, 4.95546601016601073700922742429, 5.85709392413878761029986858692, 6.80839574406470778379936640531, 7.43573288921343419416508140460, 7.961101993895220164198444168481, 9.214504908234327098902604345639
