| L(s) = 1 | − 3-s + 4.01·5-s − 1.63·7-s + 9-s + 1.79·11-s − 2.87·13-s − 4.01·15-s + 4.87·17-s − 3.79·19-s + 1.63·21-s − 1.79·23-s + 11.1·25-s − 27-s − 7.95·29-s − 0.570·31-s − 1.79·33-s − 6.55·35-s − 8.69·37-s + 2.87·39-s − 4.95·41-s − 10.6·43-s + 4.01·45-s + 9.38·47-s − 4.33·49-s − 4.87·51-s − 10.9·53-s + 7.21·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.79·5-s − 0.616·7-s + 0.333·9-s + 0.542·11-s − 0.797·13-s − 1.03·15-s + 1.18·17-s − 0.871·19-s + 0.355·21-s − 0.374·23-s + 2.22·25-s − 0.192·27-s − 1.47·29-s − 0.102·31-s − 0.312·33-s − 1.10·35-s − 1.42·37-s + 0.460·39-s − 0.773·41-s − 1.63·43-s + 0.598·45-s + 1.36·47-s − 0.619·49-s − 0.682·51-s − 1.50·53-s + 0.973·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 - 4.01T + 5T^{2} \) |
| 7 | \( 1 + 1.63T + 7T^{2} \) |
| 11 | \( 1 - 1.79T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 - 4.87T + 17T^{2} \) |
| 19 | \( 1 + 3.79T + 19T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 + 7.95T + 29T^{2} \) |
| 31 | \( 1 + 0.570T + 31T^{2} \) |
| 37 | \( 1 + 8.69T + 37T^{2} \) |
| 41 | \( 1 + 4.95T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 9.38T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 11.6T + 59T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 - 4.87T + 67T^{2} \) |
| 71 | \( 1 - 0.662T + 71T^{2} \) |
| 73 | \( 1 - 8.97T + 73T^{2} \) |
| 79 | \( 1 + 6.95T + 79T^{2} \) |
| 83 | \( 1 - 0.612T + 83T^{2} \) |
| 89 | \( 1 + 8.64T + 89T^{2} \) |
| 97 | \( 1 + 9.10T + 97T^{2} \) |
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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
−7.20829286468135364478845253272, −6.61185379591206169846546625662, −6.13602186761454165139122316945, −5.39607004077432935166628224150, −5.05128842683093640256720225432, −3.87593061933093898498536378537, −3.03491549210953838937818667781, −2.00789042478887717215826776310, −1.46653185287160632289353459408, 0,
1.46653185287160632289353459408, 2.00789042478887717215826776310, 3.03491549210953838937818667781, 3.87593061933093898498536378537, 5.05128842683093640256720225432, 5.39607004077432935166628224150, 6.13602186761454165139122316945, 6.61185379591206169846546625662, 7.20829286468135364478845253272
