| L(s) = 1 | − 2.45·3-s + 0.874·5-s + 3.01·9-s − 4.95·11-s − 4.52·13-s − 2.14·15-s − 17-s + 7.83·19-s + 3.18·23-s − 4.23·25-s − 0.0480·27-s − 3.17·29-s + 3.38·31-s + 12.1·33-s + 5.12·37-s + 11.0·39-s + 5.61·41-s + 11.3·43-s + 2.64·45-s − 8.59·47-s + 2.45·51-s − 14.1·53-s − 4.33·55-s − 19.2·57-s + 7.17·59-s + 6.17·61-s − 3.95·65-s + ⋯ |
| L(s) = 1 | − 1.41·3-s + 0.391·5-s + 1.00·9-s − 1.49·11-s − 1.25·13-s − 0.553·15-s − 0.242·17-s + 1.79·19-s + 0.664·23-s − 0.847·25-s − 0.00925·27-s − 0.589·29-s + 0.608·31-s + 2.11·33-s + 0.842·37-s + 1.77·39-s + 0.877·41-s + 1.73·43-s + 0.393·45-s − 1.25·47-s + 0.343·51-s − 1.93·53-s − 0.584·55-s − 2.54·57-s + 0.933·59-s + 0.790·61-s − 0.490·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 - 0.874T + 5T^{2} \) |
| 11 | \( 1 + 4.95T + 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 19 | \( 1 - 7.83T + 19T^{2} \) |
| 23 | \( 1 - 3.18T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 5.61T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 8.59T + 47T^{2} \) |
| 53 | \( 1 + 14.1T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 - 6.17T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 9.61T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 + 9.00T + 79T^{2} \) |
| 83 | \( 1 - 8.99T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 - 4.18T + 97T^{2} \) |
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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
−7.59981566345255587575520567463, −6.88507855726085637652727216996, −6.06222197928801711751397340542, −5.31457196026062554186364115946, −5.18603927446848653927592965571, −4.31878683694941275467850079585, −3.00843793426527520865929864452, −2.32512117035693073132237664348, −1.00064144077974133175013322389, 0,
1.00064144077974133175013322389, 2.32512117035693073132237664348, 3.00843793426527520865929864452, 4.31878683694941275467850079585, 5.18603927446848653927592965571, 5.31457196026062554186364115946, 6.06222197928801711751397340542, 6.88507855726085637652727216996, 7.59981566345255587575520567463
