| L(s) = 1 | − 3-s − 5·7-s + 9-s + 6·11-s + 3·13-s − 2·17-s − 19-s + 5·21-s − 2·23-s − 27-s − 6·29-s + 3·31-s − 6·33-s + 6·37-s − 3·39-s + 4·41-s − 11·43-s − 10·47-s + 18·49-s + 2·51-s + 8·53-s + 57-s + 6·59-s − 3·61-s − 5·63-s + 67-s + 2·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.88·7-s + 1/3·9-s + 1.80·11-s + 0.832·13-s − 0.485·17-s − 0.229·19-s + 1.09·21-s − 0.417·23-s − 0.192·27-s − 1.11·29-s + 0.538·31-s − 1.04·33-s + 0.986·37-s − 0.480·39-s + 0.624·41-s − 1.67·43-s − 1.45·47-s + 18/7·49-s + 0.280·51-s + 1.09·53-s + 0.132·57-s + 0.781·59-s − 0.384·61-s − 0.629·63-s + 0.122·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.147841951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147841951\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.470755379116205719650686137329, −7.25015765838153934650114982407, −6.57866594890582966978496415661, −6.29525910113004546443291173899, −5.65957706191857549660386964066, −4.34016250963614048822652790348, −3.80202053926083485856555514700, −3.09953565185140825305180386048, −1.77018928852372603591016942813, −0.61135458919343851602623923330,
0.61135458919343851602623923330, 1.77018928852372603591016942813, 3.09953565185140825305180386048, 3.80202053926083485856555514700, 4.34016250963614048822652790348, 5.65957706191857549660386964066, 6.29525910113004546443291173899, 6.57866594890582966978496415661, 7.25015765838153934650114982407, 8.470755379116205719650686137329
