| L(s) = 1 | − 1.16·2-s − 1.28·3-s − 0.648·4-s + 5-s + 1.48·6-s − 0.641·7-s + 3.07·8-s − 1.35·9-s − 1.16·10-s + 11-s + 0.830·12-s − 2.10·13-s + 0.746·14-s − 1.28·15-s − 2.28·16-s + 0.343·17-s + 1.58·18-s − 5.83·19-s − 0.648·20-s + 0.822·21-s − 1.16·22-s − 6.18·23-s − 3.94·24-s + 25-s + 2.45·26-s + 5.58·27-s + 0.416·28-s + ⋯ |
| L(s) = 1 | − 0.822·2-s − 0.739·3-s − 0.324·4-s + 0.447·5-s + 0.607·6-s − 0.242·7-s + 1.08·8-s − 0.453·9-s − 0.367·10-s + 0.301·11-s + 0.239·12-s − 0.584·13-s + 0.199·14-s − 0.330·15-s − 0.570·16-s + 0.0833·17-s + 0.372·18-s − 1.33·19-s − 0.145·20-s + 0.179·21-s − 0.247·22-s − 1.28·23-s − 0.804·24-s + 0.200·25-s + 0.480·26-s + 1.07·27-s + 0.0786·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4505039786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4505039786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + 1.16T + 2T^{2} \) |
| 3 | \( 1 + 1.28T + 3T^{2} \) |
| 7 | \( 1 + 0.641T + 7T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 17 | \( 1 - 0.343T + 17T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 23 | \( 1 + 6.18T + 23T^{2} \) |
| 29 | \( 1 - 7.74T + 29T^{2} \) |
| 31 | \( 1 - 2.91T + 31T^{2} \) |
| 37 | \( 1 - 0.493T + 37T^{2} \) |
| 41 | \( 1 + 8.13T + 41T^{2} \) |
| 43 | \( 1 - 3.22T + 43T^{2} \) |
| 47 | \( 1 + 8.99T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 7.45T + 61T^{2} \) |
| 67 | \( 1 + 2.50T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 79 | \( 1 - 0.783T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 2.69T + 89T^{2} \) |
| 97 | \( 1 - 9.65T + 97T^{2} \) |
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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
−8.362057910701691654171140613558, −8.086160816945672338053589075726, −6.78842696302494737292210678651, −6.42741843991255259457457628143, −5.49412344799903452039071202155, −4.77743134240370962026984202860, −4.03318949671884488923643969010, −2.73635117909004920860910432081, −1.69303147812491358566666725224, −0.45366689420985676690791353788,
0.45366689420985676690791353788, 1.69303147812491358566666725224, 2.73635117909004920860910432081, 4.03318949671884488923643969010, 4.77743134240370962026984202860, 5.49412344799903452039071202155, 6.42741843991255259457457628143, 6.78842696302494737292210678651, 8.086160816945672338053589075726, 8.362057910701691654171140613558
