| L(s) = 1 | + 3-s + 4·7-s + 9-s + 2·13-s − 6·17-s + 4·19-s + 4·21-s + 27-s + 6·29-s + 8·31-s + 2·37-s + 2·39-s − 6·41-s − 4·43-s + 9·49-s − 6·51-s − 6·53-s + 4·57-s + 10·61-s + 4·63-s − 4·67-s − 2·73-s + 8·79-s + 81-s + 12·83-s + 6·87-s + 18·89-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.872·21-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 0.529·57-s + 1.28·61-s + 0.503·63-s − 0.488·67-s − 0.234·73-s + 0.900·79-s + 1/9·81-s + 1.31·83-s + 0.643·87-s + 1.90·89-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
| \(L(1)\) |
\(\approx\) |
\(3.179937329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.179937329\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−8.286230572976717874608236616986, −7.80207005021400136120224531772, −6.87333704174627289182797723156, −6.23991034800022982622767523992, −5.03421875022587082259817541459, −4.69263356392122883726033030816, −3.79769619454667096438258605845, −2.77580512142874145203019878589, −1.92677095659654979130865508090, −1.02898483249670331178782515928,
1.02898483249670331178782515928, 1.92677095659654979130865508090, 2.77580512142874145203019878589, 3.79769619454667096438258605845, 4.69263356392122883726033030816, 5.03421875022587082259817541459, 6.23991034800022982622767523992, 6.87333704174627289182797723156, 7.80207005021400136120224531772, 8.286230572976717874608236616986
