| L(s) = 1 | + 2-s + 4-s − 3.18·5-s + 8-s − 3.18·10-s − 3.18·11-s + 5.70·13-s + 16-s + 1.52·17-s − 1.28·19-s − 3.18·20-s − 3.18·22-s − 2.23·23-s + 5.12·25-s + 5.70·26-s − 7.08·29-s + 9.42·31-s + 32-s + 1.52·34-s − 37-s − 1.28·38-s − 3.18·40-s + 5.60·41-s − 6.82·43-s − 3.18·44-s − 2.23·46-s − 5.82·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.42·5-s + 0.353·8-s − 1.00·10-s − 0.959·11-s + 1.58·13-s + 0.250·16-s + 0.369·17-s − 0.294·19-s − 0.711·20-s − 0.678·22-s − 0.466·23-s + 1.02·25-s + 1.11·26-s − 1.31·29-s + 1.69·31-s + 0.176·32-s + 0.260·34-s − 0.164·37-s − 0.208·38-s − 0.503·40-s + 0.875·41-s − 1.04·43-s − 0.479·44-s − 0.330·46-s − 0.850·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.145941184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.145941184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.18T + 5T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 - 1.52T + 17T^{2} \) |
| 19 | \( 1 + 1.28T + 19T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 + 7.08T + 29T^{2} \) |
| 31 | \( 1 - 9.42T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 5.60T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 - 2.05T + 53T^{2} \) |
| 59 | \( 1 + 1.12T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 + 8.69T + 71T^{2} \) |
| 73 | \( 1 + 4.96T + 73T^{2} \) |
| 79 | \( 1 + 4.13T + 79T^{2} \) |
| 83 | \( 1 - 8.06T + 83T^{2} \) |
| 89 | \( 1 + 0.225T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−7.88671683675165328711663580537, −7.20822252546250757555036289325, −6.33740288308311858933409534876, −5.78351694833849275374795947064, −4.87712409142002815459354333580, −4.23526429598147703484025508047, −3.55505180631878830641052110990, −3.05108018833996235682809206254, −1.88160986687871992399170087813, −0.64539924975442040367559036892,
0.64539924975442040367559036892, 1.88160986687871992399170087813, 3.05108018833996235682809206254, 3.55505180631878830641052110990, 4.23526429598147703484025508047, 4.87712409142002815459354333580, 5.78351694833849275374795947064, 6.33740288308311858933409534876, 7.20822252546250757555036289325, 7.88671683675165328711663580537
