| L(s) = 1 | + 4.47·7-s − 3·9-s − 2·11-s − 4.47·13-s + 6·19-s − 4.47·23-s − 4.47·37-s − 2·41-s + 13.4·47-s + 13.0·49-s − 13.4·53-s − 14·59-s − 13.4·63-s − 8.94·77-s + 9·81-s − 14·89-s − 20.0·91-s + 6·99-s − 4.47·103-s + 13.4·117-s + ⋯ |
| L(s) = 1 | + 1.69·7-s − 9-s − 0.603·11-s − 1.24·13-s + 1.37·19-s − 0.932·23-s − 0.735·37-s − 0.312·41-s + 1.95·47-s + 1.85·49-s − 1.84·53-s − 1.82·59-s − 1.69·63-s − 1.01·77-s + 81-s − 1.48·89-s − 2.09·91-s + 0.603·99-s − 0.440·103-s + 1.24·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 3T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.75229516030121008740640202959, −7.26969093873281983373662725382, −6.10213792461213744027679911261, −5.26216740694083247746134469576, −5.06517884846806027744058186803, −4.15656857014323267417185756525, −3.02093624972826930217720233343, −2.32378814973219626397608918861, −1.40418623019367695513879974830, 0,
1.40418623019367695513879974830, 2.32378814973219626397608918861, 3.02093624972826930217720233343, 4.15656857014323267417185756525, 5.06517884846806027744058186803, 5.26216740694083247746134469576, 6.10213792461213744027679911261, 7.26969093873281983373662725382, 7.75229516030121008740640202959
