| L(s) = 1 | − 2-s − 1.52·3-s + 4-s + 1.45·5-s + 1.52·6-s − 7-s − 8-s − 0.678·9-s − 1.45·10-s − 1.65·11-s − 1.52·12-s + 14-s − 2.21·15-s + 16-s + 3.46·17-s + 0.678·18-s + 3.33·19-s + 1.45·20-s + 1.52·21-s + 1.65·22-s − 1.21·23-s + 1.52·24-s − 2.88·25-s + 5.60·27-s − 28-s + 0.0291·29-s + 2.21·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.879·3-s + 0.5·4-s + 0.650·5-s + 0.622·6-s − 0.377·7-s − 0.353·8-s − 0.226·9-s − 0.460·10-s − 0.498·11-s − 0.439·12-s + 0.267·14-s − 0.572·15-s + 0.250·16-s + 0.839·17-s + 0.159·18-s + 0.766·19-s + 0.325·20-s + 0.332·21-s + 0.352·22-s − 0.253·23-s + 0.311·24-s − 0.576·25-s + 1.07·27-s − 0.188·28-s + 0.00542·29-s + 0.404·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 1.52T + 3T^{2} \) |
| 5 | \( 1 - 1.45T + 5T^{2} \) |
| 11 | \( 1 + 1.65T + 11T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 1.21T + 23T^{2} \) |
| 29 | \( 1 - 0.0291T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 + 9.25T + 37T^{2} \) |
| 41 | \( 1 + 5.96T + 41T^{2} \) |
| 43 | \( 1 - 9.27T + 43T^{2} \) |
| 47 | \( 1 - 1.91T + 47T^{2} \) |
| 53 | \( 1 - 1.03T + 53T^{2} \) |
| 59 | \( 1 - 6.20T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 + 6.69T + 71T^{2} \) |
| 73 | \( 1 - 1.28T + 73T^{2} \) |
| 79 | \( 1 - 9.59T + 79T^{2} \) |
| 83 | \( 1 + 0.831T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 0.856T + 97T^{2} \) |
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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
−8.712851527069101022006066767475, −7.78570627978228757510889651335, −7.08100902413002090349315289180, −6.10347095988923232326619711657, −5.70352133581664531618313007206, −4.92659480711421367914707791398, −3.48258647498979438370433432366, −2.53934972940522784851077216081, −1.29857025654858018418485546464, 0,
1.29857025654858018418485546464, 2.53934972940522784851077216081, 3.48258647498979438370433432366, 4.92659480711421367914707791398, 5.70352133581664531618313007206, 6.10347095988923232326619711657, 7.08100902413002090349315289180, 7.78570627978228757510889651335, 8.712851527069101022006066767475
