| L(s) = 1 | + i·3-s − 2.52i·5-s + 2.52·7-s − 9-s + 2.37i·11-s + 2.52·15-s + 0.372·17-s + i·19-s + 2.52i·21-s + 5.04·23-s − 1.37·25-s − i·27-s + 1.58i·29-s + 3.46·31-s − 2.37·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.12i·5-s + 0.954·7-s − 0.333·9-s + 0.715i·11-s + 0.651·15-s + 0.0902·17-s + 0.229i·19-s + 0.550i·21-s + 1.05·23-s − 0.274·25-s − 0.192i·27-s + 0.294i·29-s + 0.622·31-s − 0.412·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.166937788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.166937788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 + 2.52iT - 5T^{2} \) |
| 7 | \( 1 - 2.52T + 7T^{2} \) |
| 11 | \( 1 - 2.37iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 0.372T + 17T^{2} \) |
| 23 | \( 1 - 5.04T + 23T^{2} \) |
| 29 | \( 1 - 1.58iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 3.16iT - 37T^{2} \) |
| 41 | \( 1 + 2.74T + 41T^{2} \) |
| 43 | \( 1 - 11.1iT - 43T^{2} \) |
| 47 | \( 1 - 2.81T + 47T^{2} \) |
| 53 | \( 1 - 6.63iT - 53T^{2} \) |
| 59 | \( 1 + 8.74iT - 59T^{2} \) |
| 61 | \( 1 - 2.81iT - 61T^{2} \) |
| 67 | \( 1 - 0.744iT - 67T^{2} \) |
| 71 | \( 1 - 8.80T + 71T^{2} \) |
| 73 | \( 1 - 7.62T + 73T^{2} \) |
| 79 | \( 1 + 6.63T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 - 2.74T + 97T^{2} \) |
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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
−8.594139328703938342759368268393, −8.012384589668687199936287456852, −7.24836920964554077201222005628, −6.20900015242647701940121823400, −5.21463135700854276144055369811, −4.80570549778789383847832675826, −4.24737565364969576326831512593, −3.10700409934042296184544773813, −1.89861075128541886946748071074, −0.954466798946650230012943703097,
0.840803286581056185048502976721, 2.03777930032659113687010860944, 2.87451640959107191619539787576, 3.65668561422185189362016376081, 4.83375526166666436281251458631, 5.55936507313161851733452084592, 6.49214432953878573982386550844, 6.97529555939808193947845860067, 7.73067898956387495281043789430, 8.390388441056419064417612479458
