| L(s) = 1 | + 1.70·3-s + 2.63·7-s − 0.0783·9-s + 5.41·11-s + 6.34·13-s − 3.41·17-s + 3.26·19-s + 4.49·21-s + 1.36·23-s − 5.26·27-s − 2·29-s + 4.68·31-s + 9.26·33-s − 5.75·37-s + 10.8·39-s − 7.75·41-s − 4.44·43-s + 4.78·47-s − 0.0783·49-s − 5.84·51-s + 1.65·53-s + 5.57·57-s − 3.26·59-s + 2.49·61-s − 0.206·63-s − 7.86·67-s + 2.34·69-s + ⋯ |
| L(s) = 1 | + 0.986·3-s + 0.994·7-s − 0.0261·9-s + 1.63·11-s + 1.75·13-s − 0.829·17-s + 0.748·19-s + 0.981·21-s + 0.285·23-s − 1.01·27-s − 0.371·29-s + 0.840·31-s + 1.61·33-s − 0.946·37-s + 1.73·39-s − 1.21·41-s − 0.678·43-s + 0.698·47-s − 0.0111·49-s − 0.818·51-s + 0.227·53-s + 0.738·57-s − 0.424·59-s + 0.319·61-s − 0.0259·63-s − 0.960·67-s + 0.281·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.493808784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.493808784\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.70T + 3T^{2} \) |
| 7 | \( 1 - 2.63T + 7T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 17 | \( 1 + 3.41T + 17T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 - 1.36T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 + 5.75T + 37T^{2} \) |
| 41 | \( 1 + 7.75T + 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 - 4.78T + 47T^{2} \) |
| 53 | \( 1 - 1.65T + 53T^{2} \) |
| 59 | \( 1 + 3.26T + 59T^{2} \) |
| 61 | \( 1 - 2.49T + 61T^{2} \) |
| 67 | \( 1 + 7.86T + 67T^{2} \) |
| 71 | \( 1 - 6.15T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 8.52T + 89T^{2} \) |
| 97 | \( 1 - 4.58T + 97T^{2} \) |
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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
−8.635282715005541800551636815131, −8.259433654003237704273075325312, −7.23052958930592378318656894902, −6.49567971302889179372872419451, −5.68945535701096759670973460187, −4.62421486453074319840843531418, −3.79544877020401997786366703596, −3.22728818203837251404544941036, −1.91758686782920236702047770151, −1.23641998370575971302954819884,
1.23641998370575971302954819884, 1.91758686782920236702047770151, 3.22728818203837251404544941036, 3.79544877020401997786366703596, 4.62421486453074319840843531418, 5.68945535701096759670973460187, 6.49567971302889179372872419451, 7.23052958930592378318656894902, 8.259433654003237704273075325312, 8.635282715005541800551636815131
