| L(s) = 1 | − 5-s + (−2.27 + 1.35i)7-s − 2.71i·11-s − 6.54i·13-s + 1.53·17-s + 2.30i·19-s + 3.83i·23-s + 25-s − 3.83i·29-s + 3.25i·31-s + (2.27 − 1.35i)35-s + 3.01·37-s − 6.54·41-s + 0.468·43-s + 9.11·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + (−0.858 + 0.512i)7-s − 0.817i·11-s − 1.81i·13-s + 0.371·17-s + 0.528i·19-s + 0.799i·23-s + 0.200·25-s − 0.711i·29-s + 0.584i·31-s + (0.384 − 0.229i)35-s + 0.495·37-s − 1.02·41-s + 0.0713·43-s + 1.33·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04873090890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04873090890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (2.27 - 1.35i)T \) |
| good | 11 | \( 1 + 2.71iT - 11T^{2} \) |
| 13 | \( 1 + 6.54iT - 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 19 | \( 1 - 2.30iT - 19T^{2} \) |
| 23 | \( 1 - 3.83iT - 23T^{2} \) |
| 29 | \( 1 + 3.83iT - 29T^{2} \) |
| 31 | \( 1 - 3.25iT - 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 + 6.54T + 41T^{2} \) |
| 43 | \( 1 - 0.468T + 43T^{2} \) |
| 47 | \( 1 - 9.11T + 47T^{2} \) |
| 53 | \( 1 + 9.25iT - 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 4.78iT - 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 + 2.30iT - 71T^{2} \) |
| 73 | \( 1 - 11.4iT - 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 13.0T + 83T^{2} \) |
| 89 | \( 1 + 9.60T + 89T^{2} \) |
| 97 | \( 1 - 16.9iT - 97T^{2} \) |
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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
−8.439700501067505335906163842717, −7.903467633809768085352922908522, −7.22707748749859499799391635608, −6.19210873460963259401430477528, −5.73144088553128101701826726599, −5.08249390280820857200203252406, −3.80017513768191573822661341709, −3.27620190698070969326369439585, −2.60099465243734244604956026074, −1.08453258085167055058413058885,
0.01471794133113093434815557897, 1.42262095326910749686974893102, 2.49489781641265134888818796131, 3.42627252045096291374921476713, 4.36796435372470980273445366094, 4.60860348378999867624945822462, 5.92742894228134268386635329370, 6.64311766828629207618683755151, 7.15646095520109713464666793043, 7.71405356896806648410702061292
