| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 2·13-s − 15-s − 2·17-s + 2·21-s − 2·23-s + 25-s − 27-s − 4·29-s + 4·31-s − 2·35-s + 2·37-s − 2·39-s − 4·41-s − 10·43-s + 45-s + 6·47-s − 3·49-s + 2·51-s + 6·53-s + 4·59-s + 10·61-s − 2·63-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.258·15-s − 0.485·17-s + 0.436·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 0.718·31-s − 0.338·35-s + 0.328·37-s − 0.320·39-s − 0.624·41-s − 1.52·43-s + 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s + 0.520·59-s + 1.28·61-s − 0.251·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.331363838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331363838\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.57047928672025, −12.03394653529374, −11.70882735918894, −11.16642067308246, −10.79275254244405, −10.19469763357439, −9.905592653546170, −9.526601570205239, −8.899479058297178, −8.520501529268649, −7.991972492860887, −7.397236211989159, −6.729717836414781, −6.578407999504500, −6.112636403069989, −5.476601049550071, −5.221664434492522, −4.540576318087220, −3.891021107310660, −3.611493403854203, −2.852067385008273, −2.308607093376422, −1.707303415948011, −1.040418711367283, −0.3417791676628779,
0.3417791676628779, 1.040418711367283, 1.707303415948011, 2.308607093376422, 2.852067385008273, 3.611493403854203, 3.891021107310660, 4.540576318087220, 5.221664434492522, 5.476601049550071, 6.112636403069989, 6.578407999504500, 6.729717836414781, 7.397236211989159, 7.991972492860887, 8.520501529268649, 8.899479058297178, 9.526601570205239, 9.905592653546170, 10.19469763357439, 10.79275254244405, 11.16642067308246, 11.70882735918894, 12.03394653529374, 12.57047928672025
