| L(s) = 1 | − 3·3-s − 5·5-s − 12·7-s + 9·9-s + 20·11-s − 58·13-s + 15·15-s − 70·17-s + 92·19-s + 36·21-s − 112·23-s + 25·25-s − 27·27-s + 66·29-s + 108·31-s − 60·33-s + 60·35-s − 58·37-s + 174·39-s + 66·41-s + 388·43-s − 45·45-s + 408·47-s − 199·49-s + 210·51-s + 474·53-s − 100·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.647·7-s + 1/3·9-s + 0.548·11-s − 1.23·13-s + 0.258·15-s − 0.998·17-s + 1.11·19-s + 0.374·21-s − 1.01·23-s + 1/5·25-s − 0.192·27-s + 0.422·29-s + 0.625·31-s − 0.316·33-s + 0.289·35-s − 0.257·37-s + 0.714·39-s + 0.251·41-s + 1.37·43-s − 0.149·45-s + 1.26·47-s − 0.580·49-s + 0.576·51-s + 1.22·53-s − 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.019422244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019422244\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| good | 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 + 70 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 + 58 T + p^{3} T^{2} \) |
| 41 | \( 1 - 66 T + p^{3} T^{2} \) |
| 43 | \( 1 - 388 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 - 474 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 14 T + p^{3} T^{2} \) |
| 67 | \( 1 - 276 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 + 790 T + p^{3} T^{2} \) |
| 79 | \( 1 + 308 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1210 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1426 T + p^{3} T^{2} \) |
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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
−10.53545572226521190708466661172, −9.768416494543492849583250272797, −8.928341084737447226908896731785, −7.64428948063680499239673085200, −6.90369183200259293291984861756, −5.96075398583660698174455572226, −4.81292478861330795123495123676, −3.84135656239800401581057045257, −2.44315789238495921693818629286, −0.63359106873636744825574379802,
0.63359106873636744825574379802, 2.44315789238495921693818629286, 3.84135656239800401581057045257, 4.81292478861330795123495123676, 5.96075398583660698174455572226, 6.90369183200259293291984861756, 7.64428948063680499239673085200, 8.928341084737447226908896731785, 9.768416494543492849583250272797, 10.53545572226521190708466661172
