| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 6·11-s + 4·13-s + 16-s − 6·17-s + 4·19-s + 20-s − 6·22-s + 25-s + 4·26-s + 6·29-s + 4·31-s + 32-s − 6·34-s + 8·37-s + 4·38-s + 40-s + 8·43-s − 6·44-s + 50-s + 4·52-s + 6·53-s − 6·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 1.80·11-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.223·20-s − 1.27·22-s + 1/5·25-s + 0.784·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 1.02·34-s + 1.31·37-s + 0.648·38-s + 0.158·40-s + 1.21·43-s − 0.904·44-s + 0.141·50-s + 0.554·52-s + 0.824·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.220811440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.220811440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.216671031951744745012050772335, −7.65125639972801023012823736788, −6.68320437712445879017524905701, −6.11169774518739088507500184387, −5.33849066214287215711558916586, −4.74003498838389517315128978442, −3.86214193934957711685937697357, −2.76906233337047106018858720623, −2.32705902413160280504659000466, −0.910516499689464319674261059478,
0.910516499689464319674261059478, 2.32705902413160280504659000466, 2.76906233337047106018858720623, 3.86214193934957711685937697357, 4.74003498838389517315128978442, 5.33849066214287215711558916586, 6.11169774518739088507500184387, 6.68320437712445879017524905701, 7.65125639972801023012823736788, 8.216671031951744745012050772335
