| L(s) = 1 | + 3-s + 7-s + 9-s − 3·11-s + 13-s + 7·17-s + 21-s − 7·23-s + 27-s − 4·29-s − 8·31-s − 3·33-s + 5·37-s + 39-s − 3·41-s − 8·43-s + 6·47-s − 6·49-s + 7·51-s + 11·53-s + 4·59-s + 61-s + 63-s + 12·67-s − 7·69-s + 9·71-s − 6·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 1.69·17-s + 0.218·21-s − 1.45·23-s + 0.192·27-s − 0.742·29-s − 1.43·31-s − 0.522·33-s + 0.821·37-s + 0.160·39-s − 0.468·41-s − 1.21·43-s + 0.875·47-s − 6/7·49-s + 0.980·51-s + 1.51·53-s + 0.520·59-s + 0.128·61-s + 0.125·63-s + 1.46·67-s − 0.842·69-s + 1.06·71-s − 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.655595108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.655595108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−15.94724155851651, −15.43712344034013, −14.74629042866893, −14.42057880313686, −13.87337050880879, −13.17076587822577, −12.81671392147277, −12.06990695218331, −11.58647339927029, −10.85942352304186, −10.18766686717956, −9.869021651929587, −9.144077567924246, −8.397154569991552, −7.890380366819988, −7.572528312523792, −6.782047153040547, −5.801419829981712, −5.456424503211351, −4.690788307097484, −3.675528232147463, −3.446604161102520, −2.324708555606880, −1.800076324560211, −0.6776298100520156,
0.6776298100520156, 1.800076324560211, 2.324708555606880, 3.446604161102520, 3.675528232147463, 4.690788307097484, 5.456424503211351, 5.801419829981712, 6.782047153040547, 7.572528312523792, 7.890380366819988, 8.397154569991552, 9.144077567924246, 9.869021651929587, 10.18766686717956, 10.85942352304186, 11.58647339927029, 12.06990695218331, 12.81671392147277, 13.17076587822577, 13.87337050880879, 14.42057880313686, 14.74629042866893, 15.43712344034013, 15.94724155851651
