| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 4·7-s + 8-s + 9-s − 11-s + 2·12-s − 5·13-s + 4·14-s + 16-s + 18-s − 7·19-s + 8·21-s − 22-s − 3·23-s + 2·24-s − 5·26-s − 4·27-s + 4·28-s + 3·29-s + 5·31-s + 32-s − 2·33-s + 36-s + 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.577·12-s − 1.38·13-s + 1.06·14-s + 1/4·16-s + 0.235·18-s − 1.60·19-s + 1.74·21-s − 0.213·22-s − 0.625·23-s + 0.408·24-s − 0.980·26-s − 0.769·27-s + 0.755·28-s + 0.557·29-s + 0.898·31-s + 0.176·32-s − 0.348·33-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.234912920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.234912920\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−10.86904857880407915670093553274, −9.959094196473851190415672476368, −8.792701088514811395098657781704, −8.018643107671161927067049564625, −7.49089176630476738214073905158, −6.11526403192074994261800707105, −4.84556616445283073279559168134, −4.24270731824257755894856807282, −2.72322795871370525360801282206, −2.00743214549084187894112633204,
2.00743214549084187894112633204, 2.72322795871370525360801282206, 4.24270731824257755894856807282, 4.84556616445283073279559168134, 6.11526403192074994261800707105, 7.49089176630476738214073905158, 8.018643107671161927067049564625, 8.792701088514811395098657781704, 9.959094196473851190415672476368, 10.86904857880407915670093553274
