| L(s) = 1 | + 3·3-s − 2·7-s + 6·9-s − 11-s + 4·13-s + 5·17-s − 19-s − 6·21-s + 2·23-s + 9·27-s − 8·29-s − 10·31-s − 3·33-s − 6·37-s + 12·39-s − 3·41-s − 4·43-s − 4·47-s − 3·49-s + 15·51-s + 6·53-s − 3·57-s − 8·59-s + 10·61-s − 12·63-s + 67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.755·7-s + 2·9-s − 0.301·11-s + 1.10·13-s + 1.21·17-s − 0.229·19-s − 1.30·21-s + 0.417·23-s + 1.73·27-s − 1.48·29-s − 1.79·31-s − 0.522·33-s − 0.986·37-s + 1.92·39-s − 0.468·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s + 2.10·51-s + 0.824·53-s − 0.397·57-s − 1.04·59-s + 1.28·61-s − 1.51·63-s + 0.122·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.262194877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.262194877\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−11.10895608751437250126066085179, −10.06074294395351231429620083849, −9.327488870433976181451354515274, −8.583830936329317080627601349982, −7.74030442137325160286374916308, −6.84498949692110381717428002076, −5.47172965557922596957151472916, −3.71917651349631531072059458503, −3.28752852063355895230765150692, −1.81754778427961011985492566714,
1.81754778427961011985492566714, 3.28752852063355895230765150692, 3.71917651349631531072059458503, 5.47172965557922596957151472916, 6.84498949692110381717428002076, 7.74030442137325160286374916308, 8.583830936329317080627601349982, 9.327488870433976181451354515274, 10.06074294395351231429620083849, 11.10895608751437250126066085179
