| L(s) = 1 | + 2.10·2-s − 1.80·3-s + 2.43·4-s − 3.80·6-s + 2.68·7-s + 0.906·8-s + 0.260·9-s − 0.975·11-s − 4.38·12-s − 2.65·13-s + 5.64·14-s − 2.95·16-s + 0.547·18-s + 0.676·19-s − 4.84·21-s − 2.05·22-s + 4.03·23-s − 1.63·24-s − 5.58·26-s + 4.94·27-s + 6.52·28-s + 10.0·29-s − 9.85·31-s − 8.02·32-s + 1.76·33-s + 0.632·36-s − 6.83·37-s + ⋯ |
| L(s) = 1 | + 1.48·2-s − 1.04·3-s + 1.21·4-s − 1.55·6-s + 1.01·7-s + 0.320·8-s + 0.0866·9-s − 0.294·11-s − 1.26·12-s − 0.735·13-s + 1.50·14-s − 0.738·16-s + 0.129·18-s + 0.155·19-s − 1.05·21-s − 0.437·22-s + 0.841·23-s − 0.334·24-s − 1.09·26-s + 0.952·27-s + 1.23·28-s + 1.86·29-s − 1.77·31-s − 1.41·32-s + 0.306·33-s + 0.105·36-s − 1.12·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.10T + 2T^{2} \) |
| 3 | \( 1 + 1.80T + 3T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 11 | \( 1 + 0.975T + 11T^{2} \) |
| 13 | \( 1 + 2.65T + 13T^{2} \) |
| 19 | \( 1 - 0.676T + 19T^{2} \) |
| 23 | \( 1 - 4.03T + 23T^{2} \) |
| 29 | \( 1 - 10.0T + 29T^{2} \) |
| 31 | \( 1 + 9.85T + 31T^{2} \) |
| 37 | \( 1 + 6.83T + 37T^{2} \) |
| 41 | \( 1 - 7.32T + 41T^{2} \) |
| 43 | \( 1 + 8.90T + 43T^{2} \) |
| 47 | \( 1 + 0.874T + 47T^{2} \) |
| 53 | \( 1 - 2.84T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 2.07T + 61T^{2} \) |
| 67 | \( 1 - 5.73T + 67T^{2} \) |
| 71 | \( 1 + 1.12T + 71T^{2} \) |
| 73 | \( 1 + 2.99T + 73T^{2} \) |
| 79 | \( 1 - 0.409T + 79T^{2} \) |
| 83 | \( 1 + 6.57T + 83T^{2} \) |
| 89 | \( 1 + 5.62T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.15199572042873426923719030765, −6.71692375353477261524836179701, −5.83185511581699808832932621160, −5.27518684066449789248504410538, −4.89817033771722332152238372187, −4.32896574244844023147954780882, −3.25782943229836725220958056894, −2.53863274215267220527749226992, −1.43767362153808180109493377230, 0,
1.43767362153808180109493377230, 2.53863274215267220527749226992, 3.25782943229836725220958056894, 4.32896574244844023147954780882, 4.89817033771722332152238372187, 5.27518684066449789248504410538, 5.83185511581699808832932621160, 6.71692375353477261524836179701, 7.15199572042873426923719030765
