| L(s) = 1 | − 1.36i·3-s − 3.28i·7-s + 1.13·9-s + 3.49·11-s − 3.41i·13-s − 7.46i·17-s − 3.49·19-s − 4.48·21-s − i·23-s − 5.64i·27-s + 3.46·29-s + 2.01·31-s − 4.77i·33-s + 0.511i·37-s − 4.66·39-s + ⋯ |
| L(s) = 1 | − 0.788i·3-s − 1.24i·7-s + 0.377·9-s + 1.05·11-s − 0.946i·13-s − 1.80i·17-s − 0.802·19-s − 0.978·21-s − 0.208i·23-s − 1.08i·27-s + 0.643·29-s + 0.361·31-s − 0.831i·33-s + 0.0840i·37-s − 0.746·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.998269343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.998269343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 + 1.36iT - 3T^{2} \) |
| 7 | \( 1 + 3.28iT - 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 + 3.41iT - 13T^{2} \) |
| 17 | \( 1 + 7.46iT - 17T^{2} \) |
| 19 | \( 1 + 3.49T + 19T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 - 0.511iT - 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 - 2.76iT - 43T^{2} \) |
| 47 | \( 1 + 0.889iT - 47T^{2} \) |
| 53 | \( 1 - 14.2iT - 53T^{2} \) |
| 59 | \( 1 - 4.71T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 2.30iT - 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 3.70iT - 73T^{2} \) |
| 79 | \( 1 - 7.97T + 79T^{2} \) |
| 83 | \( 1 + 9.42iT - 83T^{2} \) |
| 89 | \( 1 + 0.801T + 89T^{2} \) |
| 97 | \( 1 - 11.7iT - 97T^{2} \) |
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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
−7.81093987839980424957401288837, −7.20163155343368176122953156163, −6.77567565436282146589774208566, −6.07876369139770125343214711410, −4.89561874239381694051704116810, −4.30276057353699052292337907317, −3.40666242636455705605980594964, −2.42267628275701367947052431690, −1.21194649752925823927698847749, −0.60401100749895926573771387613,
1.53747372939932583032889710649, 2.23086571519148205443368688112, 3.58829001932230535162923107345, 4.04279696495672080756730937429, 4.84193100367740405431559616668, 5.69264044732926045272408005603, 6.47357585939424059330741152005, 6.91296719642063856492546139559, 8.312358015367547929715659195248, 8.625033872330695219617605952960
