| L(s) = 1 | − 0.287·3-s − 4.67·7-s − 2.91·9-s − 3.33·11-s + 5.04·13-s + 17-s − 5.83·19-s + 1.34·21-s + 7.54·23-s + 1.70·27-s + 5.83·29-s − 2.45·31-s + 0.956·33-s + 9.83·37-s − 1.44·39-s + 2.63·41-s + 7.04·43-s + 0.217·47-s + 14.8·49-s − 0.287·51-s − 4.52·53-s + 1.67·57-s + 2.43·59-s − 1.55·61-s + 13.6·63-s − 3.61·67-s − 2.16·69-s + ⋯ |
| L(s) = 1 | − 0.165·3-s − 1.76·7-s − 0.972·9-s − 1.00·11-s + 1.39·13-s + 0.242·17-s − 1.33·19-s + 0.293·21-s + 1.57·23-s + 0.327·27-s + 1.08·29-s − 0.441·31-s + 0.166·33-s + 1.61·37-s − 0.232·39-s + 0.411·41-s + 1.07·43-s + 0.0316·47-s + 2.11·49-s − 0.0402·51-s − 0.620·53-s + 0.222·57-s + 0.316·59-s − 0.199·61-s + 1.71·63-s − 0.441·67-s − 0.261·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9742264167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9742264167\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 0.287T + 3T^{2} \) |
| 7 | \( 1 + 4.67T + 7T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 - 5.04T + 13T^{2} \) |
| 19 | \( 1 + 5.83T + 19T^{2} \) |
| 23 | \( 1 - 7.54T + 23T^{2} \) |
| 29 | \( 1 - 5.83T + 29T^{2} \) |
| 31 | \( 1 + 2.45T + 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 - 2.63T + 41T^{2} \) |
| 43 | \( 1 - 7.04T + 43T^{2} \) |
| 47 | \( 1 - 0.217T + 47T^{2} \) |
| 53 | \( 1 + 4.52T + 53T^{2} \) |
| 59 | \( 1 - 2.43T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 + 3.61T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 8.30T + 73T^{2} \) |
| 79 | \( 1 + 1.37T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 + 4.27T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
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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
−9.196533224049650494412911356539, −8.714626607853727243328106945072, −7.81998446477364201474482594157, −6.67925095164713318776546584513, −6.17141721871087592775174027344, −5.50436884398880137657136752454, −4.25028625319178336653335348609, −3.17024458294242618199549250218, −2.64589993006267035741390432363, −0.65834964393018390452831069748,
0.65834964393018390452831069748, 2.64589993006267035741390432363, 3.17024458294242618199549250218, 4.25028625319178336653335348609, 5.50436884398880137657136752454, 6.17141721871087592775174027344, 6.67925095164713318776546584513, 7.81998446477364201474482594157, 8.714626607853727243328106945072, 9.196533224049650494412911356539
