| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 3·11-s − 5·13-s + 16-s + 6·17-s + 19-s − 20-s + 3·22-s − 3·23-s + 25-s + 5·26-s + 6·29-s + 4·31-s − 32-s − 6·34-s + 11·37-s − 38-s + 40-s + 3·41-s − 10·43-s − 3·44-s + 3·46-s + 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.904·11-s − 1.38·13-s + 1/4·16-s + 1.45·17-s + 0.229·19-s − 0.223·20-s + 0.639·22-s − 0.625·23-s + 1/5·25-s + 0.980·26-s + 1.11·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1.80·37-s − 0.162·38-s + 0.158·40-s + 0.468·41-s − 1.52·43-s − 0.452·44-s + 0.442·46-s + 0.437·47-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.85860917269174274568526041922, −7.61107136868021385878021373095, −6.73956025038270574859901475044, −5.82292171903660322043474064504, −5.07033212160442436778486344824, −4.26254748893705011616794053705, −3.03115857298080707532754834178, −2.54088076111570639796624439764, −1.18279007296947513903666475775, 0,
1.18279007296947513903666475775, 2.54088076111570639796624439764, 3.03115857298080707532754834178, 4.26254748893705011616794053705, 5.07033212160442436778486344824, 5.82292171903660322043474064504, 6.73956025038270574859901475044, 7.61107136868021385878021373095, 7.85860917269174274568526041922
