| L(s) = 1 | + 3.24·3-s + 2.56·5-s + 0.858·7-s + 7.49·9-s − 6.20·11-s − 1.41·13-s + 8.31·15-s + 3.93·17-s − 3.82·19-s + 2.78·21-s + 3.06·23-s + 1.58·25-s + 14.5·27-s + 8.48·29-s + 1.13·31-s − 20.1·33-s + 2.20·35-s + 8.49·37-s − 4.58·39-s + 5.34·43-s + 19.2·45-s + 6.65·47-s − 6.26·49-s + 12.7·51-s − 6.41·53-s − 15.9·55-s − 12.3·57-s + ⋯ |
| L(s) = 1 | + 1.87·3-s + 1.14·5-s + 0.324·7-s + 2.49·9-s − 1.87·11-s − 0.392·13-s + 2.14·15-s + 0.953·17-s − 0.877·19-s + 0.607·21-s + 0.639·23-s + 0.317·25-s + 2.80·27-s + 1.57·29-s + 0.203·31-s − 3.50·33-s + 0.372·35-s + 1.39·37-s − 0.734·39-s + 0.815·43-s + 2.86·45-s + 0.970·47-s − 0.894·49-s + 1.78·51-s − 0.881·53-s − 2.14·55-s − 1.64·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6724 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.352079647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.352079647\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 41 | \( 1 \) |
| good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 0.858T + 7T^{2} \) |
| 11 | \( 1 + 6.20T + 11T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 3.82T + 19T^{2} \) |
| 23 | \( 1 - 3.06T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 - 8.49T + 37T^{2} \) |
| 43 | \( 1 - 5.34T + 43T^{2} \) |
| 47 | \( 1 - 6.65T + 47T^{2} \) |
| 53 | \( 1 + 6.41T + 53T^{2} \) |
| 59 | \( 1 + 3.06T + 59T^{2} \) |
| 61 | \( 1 - 7.41T + 61T^{2} \) |
| 67 | \( 1 + 1.79T + 67T^{2} \) |
| 71 | \( 1 - 3.02T + 71T^{2} \) |
| 73 | \( 1 - 0.632T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 - 7.89T + 89T^{2} \) |
| 97 | \( 1 + 9.86T + 97T^{2} \) |
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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
−8.080300138351113416804570270532, −7.59727752208322979934892558779, −6.76010826476990944214778687170, −5.83768741327472535441693306632, −4.99474670608161952494673428685, −4.39027270593753717763490358459, −3.23238423637044424725660932578, −2.54778747143231244901315907242, −2.26734061779770868420806215508, −1.15205766768069528571859093686,
1.15205766768069528571859093686, 2.26734061779770868420806215508, 2.54778747143231244901315907242, 3.23238423637044424725660932578, 4.39027270593753717763490358459, 4.99474670608161952494673428685, 5.83768741327472535441693306632, 6.76010826476990944214778687170, 7.59727752208322979934892558779, 8.080300138351113416804570270532
