| L(s) = 1 | − 2.52·5-s + 2.91·7-s + 1.28·11-s + 3.28·13-s − 1.23·17-s + 1.28·19-s + 4.01·23-s + 1.37·25-s − 1.47·29-s − 2.57·31-s − 7.36·35-s − 37-s − 10.4·41-s + 4·43-s + 5.83·47-s + 1.50·49-s + 2.54·53-s − 3.25·55-s + 2.05·59-s + 14.4·61-s − 8.30·65-s + 13.2·67-s − 3.25·71-s + 3.86·73-s + 3.75·77-s − 4.78·79-s + 4.54·83-s + ⋯ |
| L(s) = 1 | − 1.12·5-s + 1.10·7-s + 0.388·11-s + 0.912·13-s − 0.299·17-s + 0.295·19-s + 0.838·23-s + 0.274·25-s − 0.274·29-s − 0.463·31-s − 1.24·35-s − 0.164·37-s − 1.62·41-s + 0.609·43-s + 0.850·47-s + 0.215·49-s + 0.349·53-s − 0.438·55-s + 0.267·59-s + 1.84·61-s − 1.02·65-s + 1.61·67-s − 0.386·71-s + 0.452·73-s + 0.428·77-s − 0.538·79-s + 0.498·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.749272352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.749272352\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 2.52T + 5T^{2} \) |
| 7 | \( 1 - 2.91T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 - 3.28T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 - 1.28T + 19T^{2} \) |
| 23 | \( 1 - 4.01T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 5.83T + 47T^{2} \) |
| 53 | \( 1 - 2.54T + 53T^{2} \) |
| 59 | \( 1 - 2.05T + 59T^{2} \) |
| 61 | \( 1 - 14.4T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 3.25T + 71T^{2} \) |
| 73 | \( 1 - 3.86T + 73T^{2} \) |
| 79 | \( 1 + 4.78T + 79T^{2} \) |
| 83 | \( 1 - 4.54T + 83T^{2} \) |
| 89 | \( 1 - 6.76T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−8.571590285770299356790764902353, −8.241881121532557987291989885376, −7.36859119271711698930199125486, −6.77604814335516858221182206136, −5.63661165364768095607951056588, −4.85944080983411173745651860957, −4.01109223502706989167886184496, −3.40038422195508452218928012879, −1.99908483700380751262289046642, −0.862161620629818326766251517482,
0.862161620629818326766251517482, 1.99908483700380751262289046642, 3.40038422195508452218928012879, 4.01109223502706989167886184496, 4.85944080983411173745651860957, 5.63661165364768095607951056588, 6.77604814335516858221182206136, 7.36859119271711698930199125486, 8.241881121532557987291989885376, 8.571590285770299356790764902353
