| L(s) = 1 | + 0.576·3-s + 3.94·5-s + 0.944·7-s − 2.66·9-s + 11-s − 1.84·13-s + 2.27·15-s + 2·17-s − 19-s + 0.544·21-s − 2.72·23-s + 10.5·25-s − 3.26·27-s + 7.54·29-s − 4.51·31-s + 0.576·33-s + 3.72·35-s + 6.61·37-s − 1.06·39-s + 11.4·41-s − 3.51·43-s − 10.5·45-s + 9.33·47-s − 6.10·49-s + 1.15·51-s + 2·53-s + 3.94·55-s + ⋯ |
| L(s) = 1 | + 0.332·3-s + 1.76·5-s + 0.356·7-s − 0.889·9-s + 0.301·11-s − 0.511·13-s + 0.587·15-s + 0.485·17-s − 0.229·19-s + 0.118·21-s − 0.567·23-s + 2.11·25-s − 0.628·27-s + 1.40·29-s − 0.811·31-s + 0.100·33-s + 0.629·35-s + 1.08·37-s − 0.170·39-s + 1.79·41-s − 0.535·43-s − 1.56·45-s + 1.36·47-s − 0.872·49-s + 0.161·51-s + 0.274·53-s + 0.531·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.016914858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.016914858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.576T + 3T^{2} \) |
| 5 | \( 1 - 3.94T + 5T^{2} \) |
| 7 | \( 1 - 0.944T + 7T^{2} \) |
| 13 | \( 1 + 1.84T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 - 7.54T + 29T^{2} \) |
| 31 | \( 1 + 4.51T + 31T^{2} \) |
| 37 | \( 1 - 6.61T + 37T^{2} \) |
| 41 | \( 1 - 11.4T + 41T^{2} \) |
| 43 | \( 1 + 3.51T + 43T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 8.85T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 3.92T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 - 6.54T + 73T^{2} \) |
| 79 | \( 1 - 3.88T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 6.23T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−8.635399507474650349813154913686, −8.084396495654945966422311945780, −7.03385027101420293118079825762, −6.22501077029561808693738757428, −5.64458934224361899390946415835, −5.02979934964389408215927178843, −3.91104710240959013538290110766, −2.63566956795162863989981164754, −2.27307116491759756972028624756, −1.06868189256490683637321359464,
1.06868189256490683637321359464, 2.27307116491759756972028624756, 2.63566956795162863989981164754, 3.91104710240959013538290110766, 5.02979934964389408215927178843, 5.64458934224361899390946415835, 6.22501077029561808693738757428, 7.03385027101420293118079825762, 8.084396495654945966422311945780, 8.635399507474650349813154913686
