| L(s) = 1 | − 0.444·3-s − 4.20·5-s − 3.82·7-s − 2.80·9-s − 1.64·11-s − 3.46·13-s + 1.86·15-s + 17-s + 3.25·19-s + 1.69·21-s + 2.78·23-s + 12.6·25-s + 2.57·27-s − 6.58·29-s + 3.54·31-s + 0.730·33-s + 16.0·35-s − 1.04·37-s + 1.53·39-s + 11.2·41-s + 8.49·43-s + 11.7·45-s + 2.50·47-s + 7.59·49-s − 0.444·51-s − 4.55·53-s + 6.91·55-s + ⋯ |
| L(s) = 1 | − 0.256·3-s − 1.87·5-s − 1.44·7-s − 0.934·9-s − 0.495·11-s − 0.960·13-s + 0.482·15-s + 0.242·17-s + 0.746·19-s + 0.370·21-s + 0.580·23-s + 2.53·25-s + 0.496·27-s − 1.22·29-s + 0.636·31-s + 0.127·33-s + 2.71·35-s − 0.171·37-s + 0.246·39-s + 1.75·41-s + 1.29·43-s + 1.75·45-s + 0.366·47-s + 1.08·49-s − 0.0622·51-s − 0.626·53-s + 0.931·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 + 0.444T + 3T^{2} \) |
| 5 | \( 1 + 4.20T + 5T^{2} \) |
| 7 | \( 1 + 3.82T + 7T^{2} \) |
| 11 | \( 1 + 1.64T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 19 | \( 1 - 3.25T + 19T^{2} \) |
| 23 | \( 1 - 2.78T + 23T^{2} \) |
| 29 | \( 1 + 6.58T + 29T^{2} \) |
| 31 | \( 1 - 3.54T + 31T^{2} \) |
| 37 | \( 1 + 1.04T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 8.49T + 43T^{2} \) |
| 47 | \( 1 - 2.50T + 47T^{2} \) |
| 53 | \( 1 + 4.55T + 53T^{2} \) |
| 61 | \( 1 + 1.90T + 61T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 + 2.08T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 6.56T + 79T^{2} \) |
| 83 | \( 1 - 3.63T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 2.50T + 97T^{2} \) |
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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
−7.56281541574343933191175018564, −6.98571757344323560721936081439, −6.10026988498156832624946051116, −5.41478379742787127154059023333, −4.57677033202846287218036484645, −3.83595310101080080464606885503, −3.03042784049252249397241224735, −2.73641665747854225913430219853, −0.71513035962373545298490423272, 0,
0.71513035962373545298490423272, 2.73641665747854225913430219853, 3.03042784049252249397241224735, 3.83595310101080080464606885503, 4.57677033202846287218036484645, 5.41478379742787127154059023333, 6.10026988498156832624946051116, 6.98571757344323560721936081439, 7.56281541574343933191175018564
