| L(s) = 1 | − 5-s + 7-s − 1.56·11-s + 6.68·13-s − 7.56·17-s − 7.12·19-s − 3.12·23-s + 25-s − 0.438·29-s + 6.24·31-s − 35-s − 8.24·37-s + 1.12·41-s − 7.12·43-s − 2.43·47-s + 49-s + 13.1·53-s + 1.56·55-s + 4·59-s − 6.87·61-s − 6.68·65-s + 2.24·67-s − 4.24·73-s − 1.56·77-s + 0.684·79-s − 12·83-s + 7.56·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 0.470·11-s + 1.85·13-s − 1.83·17-s − 1.63·19-s − 0.651·23-s + 0.200·25-s − 0.0814·29-s + 1.12·31-s − 0.169·35-s − 1.35·37-s + 0.175·41-s − 1.08·43-s − 0.355·47-s + 0.142·49-s + 1.80·53-s + 0.210·55-s + 0.520·59-s − 0.880·61-s − 0.829·65-s + 0.274·67-s − 0.496·73-s − 0.177·77-s + 0.0770·79-s − 1.31·83-s + 0.820·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 6.68T + 13T^{2} \) |
| 17 | \( 1 + 7.56T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 + 0.438T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 - 1.12T + 41T^{2} \) |
| 43 | \( 1 + 7.12T + 43T^{2} \) |
| 47 | \( 1 + 2.43T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6.87T + 61T^{2} \) |
| 67 | \( 1 - 2.24T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 - 0.684T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 5.12T + 89T^{2} \) |
| 97 | \( 1 - 1.31T + 97T^{2} \) |
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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
−8.542005645504926115697183624833, −8.036228836198284947919140686695, −6.78129836074466195756697846011, −6.41004369607984789563315397320, −5.38847663362943702159840931612, −4.32030328427404534032647059087, −3.89902830612069211860597205348, −2.60504766567610760160526252547, −1.59201670898026079357220382839, 0,
1.59201670898026079357220382839, 2.60504766567610760160526252547, 3.89902830612069211860597205348, 4.32030328427404534032647059087, 5.38847663362943702159840931612, 6.41004369607984789563315397320, 6.78129836074466195756697846011, 8.036228836198284947919140686695, 8.542005645504926115697183624833
