| L(s) = 1 | − 2.44·3-s + 3.11·5-s + 1.05·7-s + 2.97·9-s − 0.276·11-s + 3.03·13-s − 7.62·15-s − 1.15·17-s − 6.80·19-s − 2.58·21-s + 8.16·23-s + 4.71·25-s + 0.0622·27-s − 0.470·29-s − 9.46·31-s + 0.675·33-s + 3.30·35-s + 0.964·37-s − 7.42·39-s − 9.66·41-s − 4.32·43-s + 9.27·45-s + 3.94·47-s − 5.87·49-s + 2.81·51-s − 0.187·53-s − 0.861·55-s + ⋯ |
| L(s) = 1 | − 1.41·3-s + 1.39·5-s + 0.400·7-s + 0.991·9-s − 0.0833·11-s + 0.842·13-s − 1.96·15-s − 0.279·17-s − 1.56·19-s − 0.564·21-s + 1.70·23-s + 0.943·25-s + 0.0119·27-s − 0.0873·29-s − 1.70·31-s + 0.117·33-s + 0.557·35-s + 0.158·37-s − 1.18·39-s − 1.51·41-s − 0.659·43-s + 1.38·45-s + 0.575·47-s − 0.839·49-s + 0.394·51-s − 0.0257·53-s − 0.116·55-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 | \( 1 + 2.44T + 3T^{2} \) |
| 5 | \( 1 - 3.11T + 5T^{2} \) |
| 7 | \( 1 - 1.05T + 7T^{2} \) |
| 11 | \( 1 + 0.276T + 11T^{2} \) |
| 13 | \( 1 - 3.03T + 13T^{2} \) |
| 17 | \( 1 + 1.15T + 17T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 - 8.16T + 23T^{2} \) |
| 29 | \( 1 + 0.470T + 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 - 0.964T + 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 - 3.94T + 47T^{2} \) |
| 53 | \( 1 + 0.187T + 53T^{2} \) |
| 59 | \( 1 + 3.94T + 59T^{2} \) |
| 61 | \( 1 - 1.07T + 61T^{2} \) |
| 67 | \( 1 - 1.00T + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 - 0.0981T + 73T^{2} \) |
| 79 | \( 1 + 1.16T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 - 7.76T + 97T^{2} \) |
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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
−7.08917172339280297135693013262, −6.62129591138634559653327786966, −6.04388544085201813322649942617, −5.44489427510547636921033533316, −4.98350652408397560487129860216, −4.16813516633849437863705407612, −3.01734498558680054415057015013, −1.89847698245624683953938578893, −1.32656060124098838003769351930, 0,
1.32656060124098838003769351930, 1.89847698245624683953938578893, 3.01734498558680054415057015013, 4.16813516633849437863705407612, 4.98350652408397560487129860216, 5.44489427510547636921033533316, 6.04388544085201813322649942617, 6.62129591138634559653327786966, 7.08917172339280297135693013262
