| L(s) = 1 | + 3-s − 2·9-s − 3·11-s − 2·13-s + 17-s + 4·19-s − 5·27-s − 5·31-s − 3·33-s − 2·37-s − 2·39-s + 8·43-s − 6·47-s + 51-s − 6·53-s + 4·57-s + 8·61-s + 8·67-s + 4·73-s + 79-s + 81-s + 6·83-s − 3·89-s − 5·93-s + 10·97-s + 6·99-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.904·11-s − 0.554·13-s + 0.242·17-s + 0.917·19-s − 0.962·27-s − 0.898·31-s − 0.522·33-s − 0.328·37-s − 0.320·39-s + 1.21·43-s − 0.875·47-s + 0.140·51-s − 0.824·53-s + 0.529·57-s + 1.02·61-s + 0.977·67-s + 0.468·73-s + 0.112·79-s + 1/9·81-s + 0.658·83-s − 0.317·89-s − 0.518·93-s + 1.01·97-s + 0.603·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−12.82564015942650, −12.42114325813606, −11.93446951061204, −11.40728795761223, −11.04530448821580, −10.55307760590983, −10.02549766008357, −9.527717211712082, −9.244281928828672, −8.698465826224165, −8.133381802523429, −7.773771068445560, −7.492290467180241, −6.828130014201749, −6.340247934903799, −5.577456226751349, −5.356824990736886, −4.940528414077314, −4.175058481477081, −3.591746200097900, −3.166114758016815, −2.609613057531320, −2.221387796641032, −1.531772900719278, −0.6845809251792562, 0,
0.6845809251792562, 1.531772900719278, 2.221387796641032, 2.609613057531320, 3.166114758016815, 3.591746200097900, 4.175058481477081, 4.940528414077314, 5.356824990736886, 5.577456226751349, 6.340247934903799, 6.828130014201749, 7.492290467180241, 7.773771068445560, 8.133381802523429, 8.698465826224165, 9.244281928828672, 9.527717211712082, 10.02549766008357, 10.55307760590983, 11.04530448821580, 11.40728795761223, 11.93446951061204, 12.42114325813606, 12.82564015942650
