| L(s) = 1 | − 2-s + 4-s + 1.41·5-s + 1.41·7-s − 8-s − 1.41·10-s − 1.41·14-s + 16-s + 17-s + 1.41·20-s − 1.41·23-s + 1.00·25-s + 1.41·28-s − 1.41·29-s − 1.41·31-s − 32-s − 34-s + 2.00·35-s − 1.41·37-s − 1.41·40-s − 2·43-s + 1.41·46-s + 1.00·49-s − 1.00·50-s − 1.41·56-s + 1.41·58-s + 1.41·61-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 1.41·5-s + 1.41·7-s − 8-s − 1.41·10-s − 1.41·14-s + 16-s + 17-s + 1.41·20-s − 1.41·23-s + 1.00·25-s + 1.41·28-s − 1.41·29-s − 1.41·31-s − 32-s − 34-s + 2.00·35-s − 1.41·37-s − 1.41·40-s − 2·43-s + 1.41·46-s + 1.00·49-s − 1.00·50-s − 1.41·56-s + 1.41·58-s + 1.41·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9992016210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9992016210\) |
| \(L(1)\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 2T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.926047464720743482765543244786, −9.172632880674522076251535293166, −8.338234300555155476379446874638, −7.67742463348911590685979357817, −6.74610032239481169120774554765, −5.65405133874484537786649750719, −5.27787217812789025639869372087, −3.58716429165084320403398608547, −2.02353823104298509092755551481, −1.65444733736732027359488382683,
1.65444733736732027359488382683, 2.02353823104298509092755551481, 3.58716429165084320403398608547, 5.27787217812789025639869372087, 5.65405133874484537786649750719, 6.74610032239481169120774554765, 7.67742463348911590685979357817, 8.338234300555155476379446874638, 9.172632880674522076251535293166, 9.926047464720743482765543244786
