| L(s) = 1 | + 3-s − 2·4-s − 3·5-s + 7-s + 9-s − 2·12-s − 4·13-s − 3·15-s + 4·16-s − 3·17-s + 2·19-s + 6·20-s + 21-s + 4·25-s + 27-s − 2·28-s − 6·29-s + 2·31-s − 3·35-s − 2·36-s − 10·37-s − 4·39-s + 6·41-s + 11·43-s − 3·45-s − 3·47-s + 4·48-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.577·12-s − 1.10·13-s − 0.774·15-s + 16-s − 0.727·17-s + 0.458·19-s + 1.34·20-s + 0.218·21-s + 4/5·25-s + 0.192·27-s − 0.377·28-s − 1.11·29-s + 0.359·31-s − 0.507·35-s − 1/3·36-s − 1.64·37-s − 0.640·39-s + 0.937·41-s + 1.67·43-s − 0.447·45-s − 0.437·47-s + 0.577·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.007651641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007651641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.938771221369423586808013282286, −8.054015622720724393492292539861, −7.63418768439112576104807700047, −6.94874681262006083051340596099, −5.52013750562433585517140497981, −4.74201907031432468419393146023, −4.07866007264882139603057037238, −3.43360809031156373981720630144, −2.23198175859510101209654364744, −0.60850792459387121663536069315,
0.60850792459387121663536069315, 2.23198175859510101209654364744, 3.43360809031156373981720630144, 4.07866007264882139603057037238, 4.74201907031432468419393146023, 5.52013750562433585517140497981, 6.94874681262006083051340596099, 7.63418768439112576104807700047, 8.054015622720724393492292539861, 8.938771221369423586808013282286
