| L(s) = 1 | − 3·11-s + 2·13-s + 2·19-s + 7·23-s + 3·29-s − 6·31-s − 3·37-s − 5·43-s − 2·47-s − 2·53-s + 10·59-s + 8·61-s + 9·67-s + 9·71-s + 8·73-s + 79-s − 14·83-s − 6·89-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.904·11-s + 0.554·13-s + 0.458·19-s + 1.45·23-s + 0.557·29-s − 1.07·31-s − 0.493·37-s − 0.762·43-s − 0.291·47-s − 0.274·53-s + 1.30·59-s + 1.02·61-s + 1.09·67-s + 1.06·71-s + 0.936·73-s + 0.112·79-s − 1.53·83-s − 0.635·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.410937462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.410937462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.08352581776165, −12.81565266277793, −12.37585946497163, −11.64312121341977, −11.22923056152455, −10.93446459480947, −10.33548567947324, −9.893310786977586, −9.412560992906566, −8.815627285731224, −8.387401385422060, −7.987493937517346, −7.338277469725600, −6.835899763630247, −6.560410017097431, −5.585963977711269, −5.401670610777233, −4.906939512526836, −4.232360159930589, −3.521551179487711, −3.180765395983914, −2.489244973549899, −1.893489583059482, −1.113381825825918, −0.4884225828116153,
0.4884225828116153, 1.113381825825918, 1.893489583059482, 2.489244973549899, 3.180765395983914, 3.521551179487711, 4.232360159930589, 4.906939512526836, 5.401670610777233, 5.585963977711269, 6.560410017097431, 6.835899763630247, 7.338277469725600, 7.987493937517346, 8.387401385422060, 8.815627285731224, 9.412560992906566, 9.893310786977586, 10.33548567947324, 10.93446459480947, 11.22923056152455, 11.64312121341977, 12.37585946497163, 12.81565266277793, 13.08352581776165
