| L(s) = 1 | − 2.26·3-s − 5-s − 3.39·7-s + 2.10·9-s − 3.74·11-s − 4.09·13-s + 2.26·15-s + 1.81·17-s − 0.507·19-s + 7.66·21-s + 2.07·23-s + 25-s + 2.01·27-s − 2.08·29-s − 3.75·31-s + 8.46·33-s + 3.39·35-s + 5.62·37-s + 9.26·39-s − 1.53·41-s + 6.39·43-s − 2.10·45-s + 11.8·47-s + 4.50·49-s − 4.11·51-s − 11.1·53-s + 3.74·55-s + ⋯ |
| L(s) = 1 | − 1.30·3-s − 0.447·5-s − 1.28·7-s + 0.702·9-s − 1.12·11-s − 1.13·13-s + 0.583·15-s + 0.441·17-s − 0.116·19-s + 1.67·21-s + 0.432·23-s + 0.200·25-s + 0.388·27-s − 0.386·29-s − 0.674·31-s + 1.47·33-s + 0.573·35-s + 0.925·37-s + 1.48·39-s − 0.240·41-s + 0.975·43-s − 0.314·45-s + 1.72·47-s + 0.643·49-s − 0.575·51-s − 1.53·53-s + 0.505·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 + 3.74T + 11T^{2} \) |
| 13 | \( 1 + 4.09T + 13T^{2} \) |
| 17 | \( 1 - 1.81T + 17T^{2} \) |
| 19 | \( 1 + 0.507T + 19T^{2} \) |
| 23 | \( 1 - 2.07T + 23T^{2} \) |
| 29 | \( 1 + 2.08T + 29T^{2} \) |
| 31 | \( 1 + 3.75T + 31T^{2} \) |
| 37 | \( 1 - 5.62T + 37T^{2} \) |
| 41 | \( 1 + 1.53T + 41T^{2} \) |
| 43 | \( 1 - 6.39T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 3.22T + 61T^{2} \) |
| 67 | \( 1 - 2.15T + 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 + 5.04T + 73T^{2} \) |
| 79 | \( 1 - 0.358T + 79T^{2} \) |
| 83 | \( 1 - 15.6T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 + 6.27T + 97T^{2} \) |
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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
−7.35044221899181594861941143076, −6.77505758534341811974605545390, −6.01146540072154150533661781873, −5.44841248820438626075118568870, −4.88152318680249468657381887085, −4.00496766313765698748219386147, −3.06662445680496382000994169766, −2.37961986405156061258014887581, −0.74183695435515405195452736787, 0,
0.74183695435515405195452736787, 2.37961986405156061258014887581, 3.06662445680496382000994169766, 4.00496766313765698748219386147, 4.88152318680249468657381887085, 5.44841248820438626075118568870, 6.01146540072154150533661781873, 6.77505758534341811974605545390, 7.35044221899181594861941143076
