| L(s) = 1 | − 3-s + 3·7-s + 9-s + 3·11-s − 13-s − 7·17-s + 8·19-s − 3·21-s + 4·23-s − 27-s − 3·29-s + 11·31-s − 3·33-s + 39-s − 2·41-s − 8·43-s − 9·47-s + 2·49-s + 7·51-s − 9·53-s − 8·57-s + 9·59-s + 61-s + 3·63-s + 5·67-s − 4·69-s + 12·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 1.69·17-s + 1.83·19-s − 0.654·21-s + 0.834·23-s − 0.192·27-s − 0.557·29-s + 1.97·31-s − 0.522·33-s + 0.160·39-s − 0.312·41-s − 1.21·43-s − 1.31·47-s + 2/7·49-s + 0.980·51-s − 1.23·53-s − 1.05·57-s + 1.17·59-s + 0.128·61-s + 0.377·63-s + 0.610·67-s − 0.481·69-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.419586782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.419586782\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.20790738356953, −14.52551548375905, −14.06067384987697, −13.50817773843415, −13.05914763346936, −12.27287018601085, −11.67246248165022, −11.41064196833652, −11.12387944524712, −10.28817511650346, −9.648973761534161, −9.276072103488369, −8.451369138717322, −8.092826654524602, −7.318027829036732, −6.707166968950704, −6.411999594517211, −5.441276562925863, −4.836153732505320, −4.672188257253620, −3.746676549502182, −3.012331015833255, −2.053478930821129, −1.408512980975595, −0.6584186589835133,
0.6584186589835133, 1.408512980975595, 2.053478930821129, 3.012331015833255, 3.746676549502182, 4.672188257253620, 4.836153732505320, 5.441276562925863, 6.411999594517211, 6.707166968950704, 7.318027829036732, 8.092826654524602, 8.451369138717322, 9.276072103488369, 9.648973761534161, 10.28817511650346, 11.12387944524712, 11.41064196833652, 11.67246248165022, 12.27287018601085, 13.05914763346936, 13.50817773843415, 14.06067384987697, 14.52551548375905, 15.20790738356953
