| L(s) = 1 | + (2 − i)5-s + i·7-s + 3·11-s + i·13-s − 5i·17-s − 8·19-s + 2i·23-s + (3 − 4i)25-s − 29-s + 2·31-s + (1 + 2i)35-s − 10i·37-s + 6·41-s + 4i·43-s − 11i·47-s + ⋯ |
| L(s) = 1 | + (0.894 − 0.447i)5-s + 0.377i·7-s + 0.904·11-s + 0.277i·13-s − 1.21i·17-s − 1.83·19-s + 0.417i·23-s + (0.600 − 0.800i)25-s − 0.185·29-s + 0.359·31-s + (0.169 + 0.338i)35-s − 1.64i·37-s + 0.937·41-s + 0.609i·43-s − 1.60i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.187989814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.187989814\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 + i)T \) |
| 7 | \( 1 - iT \) |
| good | 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 5iT - 17T^{2} \) |
| 19 | \( 1 + 8T + 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 11iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 10T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10iT - 73T^{2} \) |
| 79 | \( 1 + 7T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 8T + 89T^{2} \) |
| 97 | \( 1 + 3iT - 97T^{2} \) |
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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
−8.311296814015353339423453055079, −7.27202359164501887283652067206, −6.55881036277127856445748667094, −5.99881895213488819165919741291, −5.20765827077458160451620783230, −4.48277900336830893324469787781, −3.66096951386659317752288171426, −2.41275913277078082374885650434, −1.88684150898391445603083759433, −0.60599157894177934534101256434,
1.16058045040668310355876228940, 2.04275632421109199330224091561, 2.93228303455092155917137751292, 3.98513766972010593433118651527, 4.51560423947223503887939278811, 5.68504872081638067948279354660, 6.32800868759938676414129164458, 6.65709230542104463692412384344, 7.61275905102633017983041275124, 8.523439373335209524878089315535
