| L(s) = 1 | + 2.82·5-s + (−1.73 + 2i)7-s + 4.89·11-s − 6·13-s − 4.89i·17-s − 8.48i·23-s + 3.00·25-s − 3.46·31-s + (−4.89 + 5.65i)35-s − 6.92i·37-s − 4.89i·41-s + 3.46·43-s + 9.79·47-s + (−1.00 − 6.92i)49-s − 9.79i·53-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + (−0.654 + 0.755i)7-s + 1.47·11-s − 1.66·13-s − 1.18i·17-s − 1.76i·23-s + 0.600·25-s − 0.622·31-s + (−0.828 + 0.956i)35-s − 1.13i·37-s − 0.765i·41-s + 0.528·43-s + 1.42·47-s + (−0.142 − 0.989i)49-s − 1.34i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.993609137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.993609137\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 6T + 13T^{2} \) |
| 17 | \( 1 + 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 4.89iT - 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 9.79iT - 53T^{2} \) |
| 59 | \( 1 + 5.65iT - 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.629012982238615208718268703029, −7.34891604156695983421391323811, −6.78235043924358531149472303833, −6.15378785425467845466616031348, −5.40439438117933763498890254740, −4.73152541907088241910197704202, −3.64429276153775966790334442467, −2.40089517523627754296147800157, −2.21674038826987141264034174774, −0.57456780150481475725401925302,
1.20817323794339089206586582713, 1.98281573854961590577186828808, 3.11127485072595814664069091424, 3.96456354441827748111798857271, 4.77500498531235740042920004200, 5.87912605466671424157332081961, 6.17368878327383644828115327333, 7.15943429689099686242668231034, 7.52962898620437959377422223180, 8.828580565525408838771160240263
