| L(s) = 1 | − 3.44·3-s + 8.89·9-s − 5.44·11-s − 1.89·17-s + 6.34·19-s − 20.3·27-s + 18.7·33-s + 6.79·41-s − 10·43-s − 7·49-s + 6.55·51-s − 21.8·57-s + 6·59-s − 0.348·67-s + 15.6·73-s + 43.4·81-s − 6.55·83-s + 4.10·89-s + 10·97-s − 48.4·99-s + 14.1·107-s − 18.7·113-s + ⋯ |
| L(s) = 1 | − 1.99·3-s + 2.96·9-s − 1.64·11-s − 0.460·17-s + 1.45·19-s − 3.91·27-s + 3.27·33-s + 1.06·41-s − 1.52·43-s − 49-s + 0.917·51-s − 2.90·57-s + 0.781·59-s − 0.0425·67-s + 1.83·73-s + 4.83·81-s − 0.719·83-s + 0.434·89-s + 1.01·97-s − 4.87·99-s + 1.36·107-s − 1.76·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3.44T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 5.44T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 1.89T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6.79T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 0.348T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6.55T + 83T^{2} \) |
| 89 | \( 1 - 4.10T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.52618259086168918070326595856, −6.84095417691906478478812419427, −6.17562070056370089250699011503, −5.32709264408628589391872086183, −5.16239684480750661246223332754, −4.36503149139871097234030572234, −3.30101000828655216554427594440, −2.08268284399654895804492999815, −0.952661276307377357255815268635, 0,
0.952661276307377357255815268635, 2.08268284399654895804492999815, 3.30101000828655216554427594440, 4.36503149139871097234030572234, 5.16239684480750661246223332754, 5.32709264408628589391872086183, 6.17562070056370089250699011503, 6.84095417691906478478812419427, 7.52618259086168918070326595856
