| L(s) = 1 | − 3-s + 3.94·5-s − 0.336·7-s + 9-s − 11-s − 13-s − 3.94·15-s + 5.60·17-s + 0.336·19-s + 0.336·21-s − 0.336·23-s + 10.5·25-s − 27-s + 4.27·29-s − 1.94·31-s + 33-s − 1.32·35-s + 4·37-s + 39-s + 2.33·41-s − 4.27·43-s + 3.94·45-s + 9.21·47-s − 6.88·49-s − 5.60·51-s − 5.21·53-s − 3.94·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.76·5-s − 0.127·7-s + 0.333·9-s − 0.301·11-s − 0.277·13-s − 1.01·15-s + 1.35·17-s + 0.0772·19-s + 0.0734·21-s − 0.0701·23-s + 2.11·25-s − 0.192·27-s + 0.794·29-s − 0.349·31-s + 0.174·33-s − 0.224·35-s + 0.657·37-s + 0.160·39-s + 0.364·41-s − 0.652·43-s + 0.587·45-s + 1.34·47-s − 0.983·49-s − 0.785·51-s − 0.716·53-s − 0.531·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1716 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.022033635\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022033635\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.94T + 5T^{2} \) |
| 7 | \( 1 + 0.336T + 7T^{2} \) |
| 17 | \( 1 - 5.60T + 17T^{2} \) |
| 19 | \( 1 - 0.336T + 19T^{2} \) |
| 23 | \( 1 + 0.336T + 23T^{2} \) |
| 29 | \( 1 - 4.27T + 29T^{2} \) |
| 31 | \( 1 + 1.94T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 2.33T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 - 9.21T + 47T^{2} \) |
| 53 | \( 1 + 5.21T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 + 2.61T + 67T^{2} \) |
| 71 | \( 1 + 0.653T + 71T^{2} \) |
| 73 | \( 1 + 0.223T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 9.83T + 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−9.649708315398521664633891364762, −8.688077167416801254125371586229, −7.63883107457308319098052302244, −6.73445814054659074315580938747, −5.93997910920493701452565675560, −5.47030428732903690284199143335, −4.64789004515614517387308575168, −3.19096523040738039634962513117, −2.17701465794419914434988468796, −1.07834343833954341185984717591,
1.07834343833954341185984717591, 2.17701465794419914434988468796, 3.19096523040738039634962513117, 4.64789004515614517387308575168, 5.47030428732903690284199143335, 5.93997910920493701452565675560, 6.73445814054659074315580938747, 7.63883107457308319098052302244, 8.688077167416801254125371586229, 9.649708315398521664633891364762
