| L(s) = 1 | + 0.175·3-s + i·5-s + (−0.684 + 2.55i)7-s − 2.96·9-s + 0.592i·11-s + 0.918i·13-s + 0.175i·15-s − 4.66i·17-s − 7.36·19-s + (−0.119 + 0.447i)21-s − 5.99i·23-s − 25-s − 1.04·27-s − 0.178·29-s + 5.80·31-s + ⋯ |
| L(s) = 1 | + 0.101·3-s + 0.447i·5-s + (−0.258 + 0.965i)7-s − 0.989·9-s + 0.178i·11-s + 0.254i·13-s + 0.0452i·15-s − 1.13i·17-s − 1.68·19-s + (−0.0261 + 0.0977i)21-s − 1.25i·23-s − 0.200·25-s − 0.201·27-s − 0.0331·29-s + 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5366005106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5366005106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.684 - 2.55i)T \) |
| good | 3 | \( 1 - 0.175T + 3T^{2} \) |
| 11 | \( 1 - 0.592iT - 11T^{2} \) |
| 13 | \( 1 - 0.918iT - 13T^{2} \) |
| 17 | \( 1 + 4.66iT - 17T^{2} \) |
| 19 | \( 1 + 7.36T + 19T^{2} \) |
| 23 | \( 1 + 5.99iT - 23T^{2} \) |
| 29 | \( 1 + 0.178T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 + 5.77T + 37T^{2} \) |
| 41 | \( 1 + 11.8iT - 41T^{2} \) |
| 43 | \( 1 - 3.02iT - 43T^{2} \) |
| 47 | \( 1 - 8.37T + 47T^{2} \) |
| 53 | \( 1 - 7.66T + 53T^{2} \) |
| 59 | \( 1 - 8.01T + 59T^{2} \) |
| 61 | \( 1 + 9.66iT - 61T^{2} \) |
| 67 | \( 1 - 3.12iT - 67T^{2} \) |
| 71 | \( 1 - 7.70iT - 71T^{2} \) |
| 73 | \( 1 + 13.7iT - 73T^{2} \) |
| 79 | \( 1 - 2.07iT - 79T^{2} \) |
| 83 | \( 1 + 5.59T + 83T^{2} \) |
| 89 | \( 1 + 4.95iT - 89T^{2} \) |
| 97 | \( 1 + 6.76iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.694344054951859618097304361677, −8.345577376345033765947351530437, −7.10938759014712995466664120716, −6.47283592792001050917503119751, −5.73218046964181050575519420127, −4.88508062076501497001339735238, −3.83231320693009309999855281438, −2.63365109372026522800920551041, −2.30252869461604457463589007628, −0.18477193530277770095371149575,
1.22820841844305307305441750108, 2.52968598441916524960307846235, 3.65690242079467289002507301785, 4.26497553791919028211541670661, 5.38390866636248118072279175040, 6.12897101719667516135056729216, 6.87696869714827333659807392818, 7.945372125903499233595736093826, 8.408751104226633409464153826607, 9.125406443940751912076358423379
