| L(s) = 1 | − 2·3-s + 7-s + 9-s − 6·17-s − 6·19-s − 2·21-s + 23-s − 5·25-s + 4·27-s − 10·29-s − 4·31-s − 2·37-s + 10·41-s − 4·43-s − 12·47-s + 49-s + 12·51-s − 6·53-s + 12·57-s + 2·59-s + 63-s − 2·69-s + 8·71-s + 6·73-s + 10·75-s − 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.45·17-s − 1.37·19-s − 0.436·21-s + 0.208·23-s − 25-s + 0.769·27-s − 1.85·29-s − 0.718·31-s − 0.328·37-s + 1.56·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s − 0.824·53-s + 1.58·57-s + 0.260·59-s + 0.125·63-s − 0.240·69-s + 0.949·71-s + 0.702·73-s + 1.15·75-s − 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 311696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 311696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.79602517209440, −12.57913879600516, −11.77873149756398, −11.27362915570696, −11.19437477269056, −10.89228761066217, −10.27189629559294, −9.662048897837781, −9.332968827533829, −8.584601427885773, −8.430364263635081, −7.738443536355170, −7.122232210687389, −6.819106816051635, −6.183320412033558, −5.863626372532895, −5.447041353179197, −4.714714204447164, −4.510488942043762, −3.888292147667142, −3.329831498737960, −2.443485812704484, −1.957228468987961, −1.518039652185260, −0.4525920594322009, 0,
0.4525920594322009, 1.518039652185260, 1.957228468987961, 2.443485812704484, 3.329831498737960, 3.888292147667142, 4.510488942043762, 4.714714204447164, 5.447041353179197, 5.863626372532895, 6.183320412033558, 6.819106816051635, 7.122232210687389, 7.738443536355170, 8.430364263635081, 8.584601427885773, 9.332968827533829, 9.662048897837781, 10.27189629559294, 10.89228761066217, 11.19437477269056, 11.27362915570696, 11.77873149756398, 12.57913879600516, 12.79602517209440
