| L(s) = 1 | − 2.85i·7-s − 1.85·11-s − 0.854i·13-s − 1.14i·17-s − 2·19-s − 4.85i·23-s + 3.70·29-s + 2.70·31-s − 5.85i·37-s + 11.5·41-s − 0.854i·43-s + 6.70i·47-s − 1.14·49-s + 4.85i·53-s + 1.14·59-s + ⋯ |
| L(s) = 1 | − 1.07i·7-s − 0.559·11-s − 0.236i·13-s − 0.277i·17-s − 0.458·19-s − 1.01i·23-s + 0.688·29-s + 0.486·31-s − 0.962i·37-s + 1.80·41-s − 0.130i·43-s + 0.978i·47-s − 0.163·49-s + 0.666i·53-s + 0.149·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.071512835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.071512835\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.85iT - 7T^{2} \) |
| 11 | \( 1 + 1.85T + 11T^{2} \) |
| 13 | \( 1 + 0.854iT - 13T^{2} \) |
| 17 | \( 1 + 1.14iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 4.85iT - 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 - 2.70T + 31T^{2} \) |
| 37 | \( 1 + 5.85iT - 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 + 0.854iT - 43T^{2} \) |
| 47 | \( 1 - 6.70iT - 47T^{2} \) |
| 53 | \( 1 - 4.85iT - 53T^{2} \) |
| 59 | \( 1 - 1.14T + 59T^{2} \) |
| 61 | \( 1 - 0.854T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + 2.70iT - 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 6.70iT - 83T^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−7.51482529440126151852041872502, −6.94589034708484319156217549490, −6.17102502679913311503688969149, −5.49248606559964020978645329771, −4.39074830398809378851194848628, −4.27854972078230859011632132517, −3.05769257015230753621171895379, −2.43335085013936468593334537583, −1.17460785403975237457566764751, −0.26839429663131308285878888309,
1.25248769020246775917348329199, 2.30764246602553307804943364337, 2.85371342426408735242344275330, 3.85423310764439988086550035443, 4.67280234771412276084942406408, 5.43190851719600368555683428732, 5.96010510916410584703821274467, 6.68033978401079489634258131509, 7.48313182401589981249680823515, 8.207424933538058890314770432190
