| L(s) = 1 | + 3.30·3-s + 1.30·5-s + 2.30·7-s + 7.90·9-s − 11-s + 0.302·13-s + 4.30·15-s + 2.60·17-s − 19-s + 7.60·21-s + 8.60·23-s − 3.30·25-s + 16.2·27-s − 4.69·29-s − 0.302·31-s − 3.30·33-s + 3·35-s − 9.21·37-s + 1.00·39-s − 6.90·41-s − 11.9·43-s + 10.3·45-s + 6·47-s − 1.69·49-s + 8.60·51-s − 3.39·53-s − 1.30·55-s + ⋯ |
| L(s) = 1 | + 1.90·3-s + 0.582·5-s + 0.870·7-s + 2.63·9-s − 0.301·11-s + 0.0839·13-s + 1.11·15-s + 0.631·17-s − 0.229·19-s + 1.65·21-s + 1.79·23-s − 0.660·25-s + 3.11·27-s − 0.872·29-s − 0.0543·31-s − 0.574·33-s + 0.507·35-s − 1.51·37-s + 0.160·39-s − 1.07·41-s − 1.81·43-s + 1.53·45-s + 0.875·47-s − 0.242·49-s + 1.20·51-s − 0.466·53-s − 0.175·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.912760794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.912760794\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.30T + 3T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 13 | \( 1 - 0.302T + 13T^{2} \) |
| 17 | \( 1 - 2.60T + 17T^{2} \) |
| 23 | \( 1 - 8.60T + 23T^{2} \) |
| 29 | \( 1 + 4.69T + 29T^{2} \) |
| 31 | \( 1 + 0.302T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 + 6.90T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 3.39T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 4.60T + 73T^{2} \) |
| 79 | \( 1 - 5.81T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 2.60T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.626909775185722067890891223447, −8.021162380852761823738338742854, −7.36868874061674564625614645022, −6.69235826602843930211618695574, −5.36292103305552409893719949211, −4.73601567000960388937364567316, −3.63894967963303536085836531378, −3.05430289660161308549977011408, −2.00742257113417754278105017779, −1.47461518742662741182570722894,
1.47461518742662741182570722894, 2.00742257113417754278105017779, 3.05430289660161308549977011408, 3.63894967963303536085836531378, 4.73601567000960388937364567316, 5.36292103305552409893719949211, 6.69235826602843930211618695574, 7.36868874061674564625614645022, 8.021162380852761823738338742854, 8.626909775185722067890891223447
