| L(s) = 1 | + 1.56·3-s + 5-s − 0.438·7-s − 0.561·9-s − 4·11-s + 1.56·13-s + 1.56·15-s + 3.56·17-s − 19-s − 0.684·21-s − 6.68·23-s + 25-s − 5.56·27-s + 7.56·29-s − 3.12·31-s − 6.24·33-s − 0.438·35-s − 7.12·37-s + 2.43·39-s − 8.24·41-s − 2·43-s − 0.561·45-s + 5.12·47-s − 6.80·49-s + 5.56·51-s − 2.43·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 0.901·3-s + 0.447·5-s − 0.165·7-s − 0.187·9-s − 1.20·11-s + 0.433·13-s + 0.403·15-s + 0.863·17-s − 0.229·19-s − 0.149·21-s − 1.39·23-s + 0.200·25-s − 1.07·27-s + 1.40·29-s − 0.560·31-s − 1.08·33-s − 0.0741·35-s − 1.17·37-s + 0.390·39-s − 1.28·41-s − 0.304·43-s − 0.0837·45-s + 0.747·47-s − 0.972·49-s + 0.778·51-s − 0.334·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 0.438T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 3.56T + 17T^{2} \) |
| 23 | \( 1 + 6.68T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 + 8.24T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 + 2.43T + 53T^{2} \) |
| 59 | \( 1 + 7.80T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 6.43T + 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 5.80T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 14T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 7.12T + 97T^{2} \) |
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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
−7.964999336848016629304640193541, −7.13000328190260261705360641600, −6.23648531246965530345979061909, −5.58519760762751725717455767764, −4.91094277981190965344838062078, −3.79366608825540564603916404990, −3.13513070501732698910189526171, −2.43059702944974876373331108624, −1.58093590209432745577214821678, 0,
1.58093590209432745577214821678, 2.43059702944974876373331108624, 3.13513070501732698910189526171, 3.79366608825540564603916404990, 4.91094277981190965344838062078, 5.58519760762751725717455767764, 6.23648531246965530345979061909, 7.13000328190260261705360641600, 7.964999336848016629304640193541
