| L(s) = 1 | − 2.30·3-s + 3.59·7-s + 2.30·9-s + 0.497·11-s − 2.64·13-s − 5.10·17-s + 0.987·19-s − 8.27·21-s + 6.41·23-s + 1.61·27-s − 5.57·29-s − 6.05·31-s − 1.14·33-s + 4.59·37-s + 6.09·39-s − 2.87·41-s − 9.48·43-s + 5.36·47-s + 5.91·49-s + 11.7·51-s + 0.307·53-s − 2.27·57-s + 1.26·59-s − 6.22·61-s + 8.26·63-s − 5.28·67-s − 14.7·69-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 1.35·7-s + 0.766·9-s + 0.150·11-s − 0.734·13-s − 1.23·17-s + 0.226·19-s − 1.80·21-s + 1.33·23-s + 0.309·27-s − 1.03·29-s − 1.08·31-s − 0.199·33-s + 0.755·37-s + 0.976·39-s − 0.448·41-s − 1.44·43-s + 0.783·47-s + 0.845·49-s + 1.64·51-s + 0.0422·53-s − 0.301·57-s + 0.164·59-s − 0.797·61-s + 1.04·63-s − 0.645·67-s − 1.77·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.119644316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.119644316\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.30T + 3T^{2} \) |
| 7 | \( 1 - 3.59T + 7T^{2} \) |
| 11 | \( 1 - 0.497T + 11T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 - 0.987T + 19T^{2} \) |
| 23 | \( 1 - 6.41T + 23T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 - 4.59T + 37T^{2} \) |
| 41 | \( 1 + 2.87T + 41T^{2} \) |
| 43 | \( 1 + 9.48T + 43T^{2} \) |
| 47 | \( 1 - 5.36T + 47T^{2} \) |
| 53 | \( 1 - 0.307T + 53T^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 + 6.22T + 61T^{2} \) |
| 67 | \( 1 + 5.28T + 67T^{2} \) |
| 71 | \( 1 - 0.151T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 0.849T + 97T^{2} \) |
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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
−7.50563823094928121492613662782, −6.91777417467528526112246879019, −6.29522713693731566063576805127, −5.39919374966804607325530938701, −4.99927704067699571287746391569, −4.56202262371237397018496416716, −3.57500523851436706556922698502, −2.35729340908865837597768515424, −1.59402329731831750047781991999, −0.54942713787176081375595469113,
0.54942713787176081375595469113, 1.59402329731831750047781991999, 2.35729340908865837597768515424, 3.57500523851436706556922698502, 4.56202262371237397018496416716, 4.99927704067699571287746391569, 5.39919374966804607325530938701, 6.29522713693731566063576805127, 6.91777417467528526112246879019, 7.50563823094928121492613662782
