| L(s) = 1 | − 3.12·3-s − 1.82·5-s + 7-s + 6.76·9-s − 11-s − 13-s + 5.69·15-s − 4.38·17-s − 0.0136·19-s − 3.12·21-s + 4.39·23-s − 1.67·25-s − 11.7·27-s + 2.69·29-s − 2.49·31-s + 3.12·33-s − 1.82·35-s − 2.23·37-s + 3.12·39-s + 6.14·41-s + 1.11·43-s − 12.3·45-s + 6.86·47-s + 49-s + 13.7·51-s + 5.30·53-s + 1.82·55-s + ⋯ |
| L(s) = 1 | − 1.80·3-s − 0.815·5-s + 0.377·7-s + 2.25·9-s − 0.301·11-s − 0.277·13-s + 1.47·15-s − 1.06·17-s − 0.00313·19-s − 0.681·21-s + 0.916·23-s − 0.335·25-s − 2.26·27-s + 0.500·29-s − 0.448·31-s + 0.543·33-s − 0.308·35-s − 0.366·37-s + 0.500·39-s + 0.959·41-s + 0.170·43-s − 1.83·45-s + 1.00·47-s + 0.142·49-s + 1.91·51-s + 0.729·53-s + 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.12T + 3T^{2} \) |
| 5 | \( 1 + 1.82T + 5T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 + 0.0136T + 19T^{2} \) |
| 23 | \( 1 - 4.39T + 23T^{2} \) |
| 29 | \( 1 - 2.69T + 29T^{2} \) |
| 31 | \( 1 + 2.49T + 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 41 | \( 1 - 6.14T + 41T^{2} \) |
| 43 | \( 1 - 1.11T + 43T^{2} \) |
| 47 | \( 1 - 6.86T + 47T^{2} \) |
| 53 | \( 1 - 5.30T + 53T^{2} \) |
| 59 | \( 1 + 7.05T + 59T^{2} \) |
| 61 | \( 1 + 3.70T + 61T^{2} \) |
| 67 | \( 1 - 4.94T + 67T^{2} \) |
| 71 | \( 1 - 7.14T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 5.87T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 4.38T + 97T^{2} \) |
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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
−7.82972233993825359662103309503, −7.22039224668935763415322058061, −6.60051216116325577535083081746, −5.80309350151782401419253581026, −5.05337842503048102679052101751, −4.51715004744661902432888768167, −3.76214167775054304225808087565, −2.31149165283672743334768990903, −0.996813965770702996713113825648, 0,
0.996813965770702996713113825648, 2.31149165283672743334768990903, 3.76214167775054304225808087565, 4.51715004744661902432888768167, 5.05337842503048102679052101751, 5.80309350151782401419253581026, 6.60051216116325577535083081746, 7.22039224668935763415322058061, 7.82972233993825359662103309503
