L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 3·9-s − 11-s − 2·13-s + 14-s + 16-s − 6·17-s − 3·18-s + 4·19-s − 22-s − 4·23-s − 2·26-s + 28-s − 2·29-s + 8·31-s + 32-s − 6·34-s − 3·36-s + 10·37-s + 4·38-s − 6·41-s − 12·43-s − 44-s − 4·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 9-s − 0.301·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.213·22-s − 0.834·23-s − 0.392·26-s + 0.188·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s − 1.02·34-s − 1/2·36-s + 1.64·37-s + 0.648·38-s − 0.937·41-s − 1.82·43-s − 0.150·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.091572992705813150503503841520, −7.36724638720596046836615184349, −6.41444614335844704712814652637, −5.92691045159899202485708703134, −4.87357292365988070618620622182, −4.59161092301521288709471744001, −3.32451610962817455478500208221, −2.66903939783330043504079877356, −1.71052969422956014629545325941, 0,
1.71052969422956014629545325941, 2.66903939783330043504079877356, 3.32451610962817455478500208221, 4.59161092301521288709471744001, 4.87357292365988070618620622182, 5.92691045159899202485708703134, 6.41444614335844704712814652637, 7.36724638720596046836615184349, 8.091572992705813150503503841520