| L(s) = 1 | − 0.837·2-s + 0.531·3-s − 1.29·4-s − 0.445·6-s − 1.15·7-s + 2.76·8-s − 2.71·9-s + 0.824·11-s − 0.689·12-s + 0.971·14-s + 0.280·16-s − 3.49·17-s + 2.27·18-s + 2.73·19-s − 0.616·21-s − 0.691·22-s + 6.44·23-s + 1.46·24-s − 3.03·27-s + 1.50·28-s + 5.08·29-s − 9.28·31-s − 5.76·32-s + 0.438·33-s + 2.93·34-s + 3.52·36-s + 6.05·37-s + ⋯ |
| L(s) = 1 | − 0.592·2-s + 0.306·3-s − 0.648·4-s − 0.181·6-s − 0.438·7-s + 0.976·8-s − 0.905·9-s + 0.248·11-s − 0.199·12-s + 0.259·14-s + 0.0701·16-s − 0.848·17-s + 0.536·18-s + 0.627·19-s − 0.134·21-s − 0.147·22-s + 1.34·23-s + 0.299·24-s − 0.584·27-s + 0.284·28-s + 0.945·29-s − 1.66·31-s − 1.01·32-s + 0.0762·33-s + 0.502·34-s + 0.587·36-s + 0.995·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.837T + 2T^{2} \) |
| 3 | \( 1 - 0.531T + 3T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 - 0.824T + 11T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 - 2.73T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 - 5.08T + 29T^{2} \) |
| 31 | \( 1 + 9.28T + 31T^{2} \) |
| 37 | \( 1 - 6.05T + 37T^{2} \) |
| 41 | \( 1 + 1.20T + 41T^{2} \) |
| 43 | \( 1 + 0.920T + 43T^{2} \) |
| 47 | \( 1 - 2.97T + 47T^{2} \) |
| 53 | \( 1 - 4.76T + 53T^{2} \) |
| 59 | \( 1 - 7.48T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 0.539T + 71T^{2} \) |
| 73 | \( 1 + 0.0375T + 73T^{2} \) |
| 79 | \( 1 - 8.77T + 79T^{2} \) |
| 83 | \( 1 + 9.33T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 16.3T + 97T^{2} \) |
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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
−8.325462353088010790784526281875, −7.38865314319950142229134553198, −6.79134486165567987233143680367, −5.74859328145398798748128634124, −5.08348434329422328433776618282, −4.17827491030921595932306247525, −3.33996090788483181344602035296, −2.47267081461661377604267884132, −1.17252740889088091279240996801, 0,
1.17252740889088091279240996801, 2.47267081461661377604267884132, 3.33996090788483181344602035296, 4.17827491030921595932306247525, 5.08348434329422328433776618282, 5.74859328145398798748128634124, 6.79134486165567987233143680367, 7.38865314319950142229134553198, 8.325462353088010790784526281875
