| L(s) = 1 | − 2·3-s + 5-s + 5·7-s + 9-s − 2·11-s − 4·13-s − 2·15-s + 3·17-s − 10·21-s − 23-s + 25-s + 4·27-s + 29-s − 31-s + 4·33-s + 5·35-s − 7·37-s + 8·39-s − 3·41-s − 4·43-s + 45-s − 12·47-s + 18·49-s − 6·51-s + 53-s − 2·55-s − 3·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s + 1.88·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.516·15-s + 0.727·17-s − 2.18·21-s − 0.208·23-s + 1/5·25-s + 0.769·27-s + 0.185·29-s − 0.179·31-s + 0.696·33-s + 0.845·35-s − 1.15·37-s + 1.28·39-s − 0.468·41-s − 0.609·43-s + 0.149·45-s − 1.75·47-s + 18/7·49-s − 0.840·51-s + 0.137·53-s − 0.269·55-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 17 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.46201125758330373730466306465, −6.92011191118727718546249621109, −5.91916756539972332610569933124, −5.23870723404427989806442302974, −5.06838785901011135898069283450, −4.33799496750345449326295313255, −3.01955596690019910027440617430, −2.02416498965557980896389449321, −1.28879573954186856296099403805, 0,
1.28879573954186856296099403805, 2.02416498965557980896389449321, 3.01955596690019910027440617430, 4.33799496750345449326295313255, 5.06838785901011135898069283450, 5.23870723404427989806442302974, 5.91916756539972332610569933124, 6.92011191118727718546249621109, 7.46201125758330373730466306465
