| L(s) = 1 | − 2.08·2-s + 2.35·4-s − 1.35·5-s + 0.351·7-s − 0.734·8-s + 2.82·10-s − 5.52·11-s − 5.17·13-s − 0.734·14-s − 3.17·16-s − 3.17·20-s + 11.5·22-s − 8.82·23-s − 3.17·25-s + 10.7·26-s + 0.827·28-s − 2.70·29-s − 0.524·31-s + 8.08·32-s − 0.475·35-s − 37-s + 0.992·40-s + 2.70·41-s − 6.52·43-s − 12.9·44-s + 18.4·46-s + 6·47-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.17·4-s − 0.604·5-s + 0.133·7-s − 0.259·8-s + 0.891·10-s − 1.66·11-s − 1.43·13-s − 0.196·14-s − 0.793·16-s − 0.710·20-s + 2.45·22-s − 1.83·23-s − 0.634·25-s + 2.11·26-s + 0.156·28-s − 0.502·29-s − 0.0941·31-s + 1.42·32-s − 0.0804·35-s − 0.164·37-s + 0.156·40-s + 0.422·41-s − 0.994·43-s − 1.95·44-s + 2.71·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1782990256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1782990256\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 7 | \( 1 - 0.351T + 7T^{2} \) |
| 11 | \( 1 + 5.52T + 11T^{2} \) |
| 13 | \( 1 + 5.17T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 + 2.70T + 29T^{2} \) |
| 31 | \( 1 + 0.524T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 + 6.52T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 4.05T + 53T^{2} \) |
| 59 | \( 1 - 5.52T + 59T^{2} \) |
| 61 | \( 1 + 1.87T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 + 5.04T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 - 7.82T + 79T^{2} \) |
| 83 | \( 1 + 8.34T + 83T^{2} \) |
| 89 | \( 1 - 4.64T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
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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
−8.422972330398766231608887127905, −7.948918416745798052111529732782, −7.58001348248452133358347317486, −6.84342452620837481203659925822, −5.65370879349837794886391780141, −4.87562099503737659412609585312, −3.92510201337969948468173452061, −2.58908043888323618132161169445, −1.95717406589227275335648745645, −0.29627147163659246408247627009,
0.29627147163659246408247627009, 1.95717406589227275335648745645, 2.58908043888323618132161169445, 3.92510201337969948468173452061, 4.87562099503737659412609585312, 5.65370879349837794886391780141, 6.84342452620837481203659925822, 7.58001348248452133358347317486, 7.948918416745798052111529732782, 8.422972330398766231608887127905
