L(s) = 1 | + 4-s + 1.24·5-s + 9-s − 1.80·13-s + 16-s + 1.24·20-s − 0.445·23-s + 0.554·25-s − 0.445·31-s + 36-s − 1.80·41-s − 1.80·43-s + 1.24·45-s + 49-s − 1.80·52-s + 64-s − 2.24·65-s − 1.80·71-s + 1.24·80-s + 81-s − 0.445·83-s − 0.445·89-s − 0.445·92-s + 0.554·100-s + 1.24·101-s + 1.24·103-s + 1.24·109-s + ⋯ |
L(s) = 1 | + 4-s + 1.24·5-s + 9-s − 1.80·13-s + 16-s + 1.24·20-s − 0.445·23-s + 0.554·25-s − 0.445·31-s + 36-s − 1.80·41-s − 1.80·43-s + 1.24·45-s + 49-s − 1.80·52-s + 64-s − 2.24·65-s − 1.80·71-s + 1.24·80-s + 81-s − 0.445·83-s − 0.445·89-s − 0.445·92-s + 0.554·100-s + 1.24·101-s + 1.24·103-s + 1.24·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.791490749\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.791490749\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2011 | \( 1+O(T) \) |
good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.24T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 1.80T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.445T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 0.445T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.80T + T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.80T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.445T + T^{2} \) |
| 89 | \( 1 + 0.445T + T^{2} \) |
| 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
−9.745431022933099867967448078128, −8.586881907403457608406430015799, −7.45881190023126099539701954613, −7.05296604711855684608828411781, −6.25947800722014296123538162185, −5.41764417430903220491527889275, −4.65219973318233909750970594027, −3.28691773354206112475033510574, −2.21394021840978122774154295075, −1.67988930515697396102481732042,
1.67988930515697396102481732042, 2.21394021840978122774154295075, 3.28691773354206112475033510574, 4.65219973318233909750970594027, 5.41764417430903220491527889275, 6.25947800722014296123538162185, 7.05296604711855684608828411781, 7.45881190023126099539701954613, 8.586881907403457608406430015799, 9.745431022933099867967448078128