| L(s) = 1 | + 3·3-s − 36·7-s + 9·9-s + 36·11-s − 54·13-s + 22·17-s − 36·19-s − 108·21-s − 144·23-s + 27·27-s + 50·29-s + 108·31-s + 108·33-s − 214·37-s − 162·39-s − 446·41-s + 252·43-s + 72·47-s + 953·49-s + 66·51-s + 22·53-s − 108·57-s + 684·59-s − 466·61-s − 324·63-s − 180·67-s − 432·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.94·7-s + 1/3·9-s + 0.986·11-s − 1.15·13-s + 0.313·17-s − 0.434·19-s − 1.12·21-s − 1.30·23-s + 0.192·27-s + 0.320·29-s + 0.625·31-s + 0.569·33-s − 0.950·37-s − 0.665·39-s − 1.69·41-s + 0.893·43-s + 0.223·47-s + 2.77·49-s + 0.181·51-s + 0.0570·53-s − 0.250·57-s + 1.50·59-s − 0.978·61-s − 0.647·63-s − 0.328·67-s − 0.753·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.453510128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.453510128\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 36 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 22 T + p^{3} T^{2} \) |
| 19 | \( 1 + 36 T + p^{3} T^{2} \) |
| 23 | \( 1 + 144 T + p^{3} T^{2} \) |
| 29 | \( 1 - 50 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 446 T + p^{3} T^{2} \) |
| 43 | \( 1 - 252 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 - 22 T + p^{3} T^{2} \) |
| 59 | \( 1 - 684 T + p^{3} T^{2} \) |
| 61 | \( 1 + 466 T + p^{3} T^{2} \) |
| 67 | \( 1 + 180 T + p^{3} T^{2} \) |
| 71 | \( 1 + 576 T + p^{3} T^{2} \) |
| 73 | \( 1 - 54 T + p^{3} T^{2} \) |
| 79 | \( 1 - 972 T + p^{3} T^{2} \) |
| 83 | \( 1 + 684 T + p^{3} T^{2} \) |
| 89 | \( 1 - 346 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1134 T + p^{3} T^{2} \) |
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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
−8.818457859415836357394367900103, −7.82514000329684952350864011715, −6.91383355030170700191875014445, −6.51188402038725023987868469784, −5.63088680876887706305236003805, −4.38354857027524054761127837734, −3.62636745302911342666207512586, −2.91709008733177270331683824381, −1.96719599498448817683326981238, −0.49423609343382783366695751830,
0.49423609343382783366695751830, 1.96719599498448817683326981238, 2.91709008733177270331683824381, 3.62636745302911342666207512586, 4.38354857027524054761127837734, 5.63088680876887706305236003805, 6.51188402038725023987868469784, 6.91383355030170700191875014445, 7.82514000329684952350864011715, 8.818457859415836357394367900103
