| L(s) = 1 | + 2·7-s − 6·13-s + 17-s − 4·19-s − 6·23-s − 5·25-s + 8·29-s − 2·31-s + 12·37-s + 2·41-s + 12·43-s + 4·47-s − 3·49-s + 14·53-s + 12·59-s − 4·61-s + 4·67-s + 6·71-s − 6·73-s + 14·79-s + 12·83-s − 6·89-s − 12·91-s − 10·97-s − 2·101-s − 8·103-s + 16·107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.66·13-s + 0.242·17-s − 0.917·19-s − 1.25·23-s − 25-s + 1.48·29-s − 0.359·31-s + 1.97·37-s + 0.312·41-s + 1.82·43-s + 0.583·47-s − 3/7·49-s + 1.92·53-s + 1.56·59-s − 0.512·61-s + 0.488·67-s + 0.712·71-s − 0.702·73-s + 1.57·79-s + 1.31·83-s − 0.635·89-s − 1.25·91-s − 1.01·97-s − 0.199·101-s − 0.788·103-s + 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.764438749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.764438749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.991337352186688545289991436091, −7.78201637450259649758483593939, −6.88696438985007832585058734279, −6.02708151781797531935060688253, −5.34109441214277489731357130863, −4.44868422960008038333800135379, −4.01961273703312551000498000103, −2.54669745357094402573494591943, −2.15339111650926817310074262135, −0.71264390768058046492344448540,
0.71264390768058046492344448540, 2.15339111650926817310074262135, 2.54669745357094402573494591943, 4.01961273703312551000498000103, 4.44868422960008038333800135379, 5.34109441214277489731357130863, 6.02708151781797531935060688253, 6.88696438985007832585058734279, 7.78201637450259649758483593939, 7.991337352186688545289991436091
