| L(s) = 1 | + 0.447·3-s + 0.901·5-s + 1.57·7-s − 2.79·9-s − 2.73·11-s − 0.127·13-s + 0.403·15-s − 0.965·17-s − 1.08·19-s + 0.706·21-s − 0.734·23-s − 4.18·25-s − 2.59·27-s + 9.81·29-s + 6.56·31-s − 1.22·33-s + 1.42·35-s − 2.91·37-s − 0.0572·39-s + 1.43·41-s + 2.61·43-s − 2.52·45-s − 2.70·47-s − 4.50·49-s − 0.432·51-s + 0.901·53-s − 2.46·55-s + ⋯ |
| L(s) = 1 | + 0.258·3-s + 0.403·5-s + 0.596·7-s − 0.933·9-s − 0.824·11-s − 0.0354·13-s + 0.104·15-s − 0.234·17-s − 0.248·19-s + 0.154·21-s − 0.153·23-s − 0.837·25-s − 0.499·27-s + 1.82·29-s + 1.17·31-s − 0.213·33-s + 0.240·35-s − 0.479·37-s − 0.00916·39-s + 0.223·41-s + 0.398·43-s − 0.376·45-s − 0.394·47-s − 0.643·49-s − 0.0604·51-s + 0.123·53-s − 0.332·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 0.447T + 3T^{2} \) |
| 5 | \( 1 - 0.901T + 5T^{2} \) |
| 7 | \( 1 - 1.57T + 7T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 + 0.127T + 13T^{2} \) |
| 17 | \( 1 + 0.965T + 17T^{2} \) |
| 19 | \( 1 + 1.08T + 19T^{2} \) |
| 23 | \( 1 + 0.734T + 23T^{2} \) |
| 29 | \( 1 - 9.81T + 29T^{2} \) |
| 31 | \( 1 - 6.56T + 31T^{2} \) |
| 37 | \( 1 + 2.91T + 37T^{2} \) |
| 41 | \( 1 - 1.43T + 41T^{2} \) |
| 43 | \( 1 - 2.61T + 43T^{2} \) |
| 47 | \( 1 + 2.70T + 47T^{2} \) |
| 53 | \( 1 - 0.901T + 53T^{2} \) |
| 59 | \( 1 + 6.31T + 59T^{2} \) |
| 61 | \( 1 + 5.52T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 - 9.68T + 71T^{2} \) |
| 73 | \( 1 + 1.43T + 73T^{2} \) |
| 79 | \( 1 - 9.80T + 79T^{2} \) |
| 83 | \( 1 + 4.88T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 + 8.63T + 97T^{2} \) |
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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
−7.76119871012325711760405027311, −6.66183052510014629699206097963, −6.13934469778991594315738722988, −5.31569306188210573666778152668, −4.80461035753557373998870521082, −3.91505954206529786726157457458, −2.82193387793391494935573333561, −2.43569672532271700121798305163, −1.35105329859266142473500319013, 0,
1.35105329859266142473500319013, 2.43569672532271700121798305163, 2.82193387793391494935573333561, 3.91505954206529786726157457458, 4.80461035753557373998870521082, 5.31569306188210573666778152668, 6.13934469778991594315738722988, 6.66183052510014629699206097963, 7.76119871012325711760405027311
