| L(s) = 1 | − 3.12·3-s + 4.24·5-s − 1.30·7-s + 6.76·9-s + 1.62·11-s − 3.29·13-s − 13.2·15-s + 1.71·17-s − 4.45·19-s + 4.08·21-s − 6.76·23-s + 13.0·25-s − 11.7·27-s + 6.13·29-s − 3.16·31-s − 5.09·33-s − 5.55·35-s − 2.83·37-s + 10.2·39-s + 0.937·43-s + 28.7·45-s + 1.88·47-s − 5.28·49-s − 5.36·51-s − 0.389·53-s + 6.91·55-s + 13.9·57-s + ⋯ |
| L(s) = 1 | − 1.80·3-s + 1.89·5-s − 0.494·7-s + 2.25·9-s + 0.491·11-s − 0.913·13-s − 3.42·15-s + 0.416·17-s − 1.02·19-s + 0.892·21-s − 1.41·23-s + 2.60·25-s − 2.26·27-s + 1.13·29-s − 0.567·31-s − 0.886·33-s − 0.938·35-s − 0.465·37-s + 1.64·39-s + 0.142·43-s + 4.27·45-s + 0.274·47-s − 0.755·49-s − 0.750·51-s − 0.0534·53-s + 0.932·55-s + 1.84·57-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
| \(L(1)\) |
\(\approx\) |
\(1.279325540\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.279325540\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.12T + 3T^{2} \) |
| 5 | \( 1 - 4.24T + 5T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 - 1.71T + 17T^{2} \) |
| 19 | \( 1 + 4.45T + 19T^{2} \) |
| 23 | \( 1 + 6.76T + 23T^{2} \) |
| 29 | \( 1 - 6.13T + 29T^{2} \) |
| 31 | \( 1 + 3.16T + 31T^{2} \) |
| 37 | \( 1 + 2.83T + 37T^{2} \) |
| 43 | \( 1 - 0.937T + 43T^{2} \) |
| 47 | \( 1 - 1.88T + 47T^{2} \) |
| 53 | \( 1 + 0.389T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 + 6.07T + 61T^{2} \) |
| 67 | \( 1 - 4.83T + 67T^{2} \) |
| 71 | \( 1 - 9.65T + 71T^{2} \) |
| 73 | \( 1 - 8.76T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 - 1.36T + 97T^{2} \) |
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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
−7.78430158016265587298165941153, −6.70064981512090239245692420611, −6.47502952633522398262152608537, −5.98293817379020888397429354753, −5.22987414331959931945363276941, −4.82863723423470630059642147952, −3.78623461981149787629695617439, −2.38759574543975738438437109243, −1.72149409478512351040594327503, −0.64238301391094371486607423248,
0.64238301391094371486607423248, 1.72149409478512351040594327503, 2.38759574543975738438437109243, 3.78623461981149787629695617439, 4.82863723423470630059642147952, 5.22987414331959931945363276941, 5.98293817379020888397429354753, 6.47502952633522398262152608537, 6.70064981512090239245692420611, 7.78430158016265587298165941153
