| L(s) = 1 | + 3-s + 4.21·5-s + 1.37·7-s + 9-s + 3.52·11-s + 2·13-s + 4.21·15-s − 6.64·17-s − 5.80·19-s + 1.37·21-s + 8·23-s + 12.7·25-s + 27-s − 9.05·29-s + 5.80·31-s + 3.52·33-s + 5.80·35-s − 3.37·37-s + 2·39-s + 10.7·41-s − 4.02·43-s + 4.21·45-s + 6.57·47-s − 5.10·49-s − 6.64·51-s − 9.39·53-s + 14.8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.88·5-s + 0.520·7-s + 0.333·9-s + 1.06·11-s + 0.554·13-s + 1.08·15-s − 1.61·17-s − 1.33·19-s + 0.300·21-s + 1.66·23-s + 2.55·25-s + 0.192·27-s − 1.68·29-s + 1.04·31-s + 0.613·33-s + 0.981·35-s − 0.555·37-s + 0.320·39-s + 1.68·41-s − 0.613·43-s + 0.628·45-s + 0.959·47-s − 0.728·49-s − 0.930·51-s − 1.29·53-s + 2.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.558893248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.558893248\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 4.21T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 11 | \( 1 - 3.52T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 6.64T + 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 9.05T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 + 3.37T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 4.02T + 43T^{2} \) |
| 47 | \( 1 - 6.57T + 47T^{2} \) |
| 53 | \( 1 + 9.39T + 53T^{2} \) |
| 59 | \( 1 - 3.05T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 + 8.04T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 7.69T + 79T^{2} \) |
| 83 | \( 1 - 14.7T + 83T^{2} \) |
| 89 | \( 1 + 6.29T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−7.947115245566634694531693547865, −6.85315062006866494661823473901, −6.49987320077356781486333119939, −5.92762422296789811980698165848, −4.94824005482558571566438195811, −4.41931166333082150945384555645, −3.42133564502810411654575998935, −2.35690546945413262201974087124, −1.93595892610575022875278059804, −1.11684699911834793097319554518,
1.11684699911834793097319554518, 1.93595892610575022875278059804, 2.35690546945413262201974087124, 3.42133564502810411654575998935, 4.41931166333082150945384555645, 4.94824005482558571566438195811, 5.92762422296789811980698165848, 6.49987320077356781486333119939, 6.85315062006866494661823473901, 7.947115245566634694531693547865
