| L(s) = 1 | + i·3-s − 4i·7-s − 9-s + 4·11-s − 6i·13-s + 2i·17-s − 4·19-s + 4·21-s − i·27-s − 10·29-s − 4·31-s + 4i·33-s − 10i·37-s + 6·39-s + 2·41-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.51i·7-s − 0.333·9-s + 1.20·11-s − 1.66i·13-s + 0.485i·17-s − 0.917·19-s + 0.872·21-s − 0.192i·27-s − 1.85·29-s − 0.718·31-s + 0.696i·33-s − 1.64i·37-s + 0.960·39-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.102449557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.102449557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 10T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.855600324213406809229411074433, −7.76606284218706126814379785198, −7.39640094616916964491836573603, −6.28176025653062753368252759041, −5.64466257177653731452203004986, −4.45842239993708755975168508801, −3.90390800618393894945947560836, −3.20718508830433167221731081658, −1.63707881080385975315184792604, −0.35822309088970059021347250441,
1.66752000241751099498930975174, 2.19483678212040870775907348656, 3.45390762867940654157333871111, 4.43920942634117379908554210131, 5.41660170813925660897631201467, 6.30336352273678372979717447891, 6.69609390625661475827684194264, 7.64500839524029593487435651870, 8.679021011543901445727968412339, 9.128956660763322960301473793717
