| L(s) = 1 | − 27·3-s − 125·5-s + 540·7-s + 729·9-s − 3.58e3·11-s + 5.99e3·13-s + 3.37e3·15-s − 2.46e4·17-s + 3.12e4·19-s − 1.45e4·21-s − 5.37e3·23-s + 1.56e4·25-s − 1.96e4·27-s − 1.94e5·29-s + 4.35e4·31-s + 9.67e4·33-s − 6.75e4·35-s − 2.44e5·37-s − 1.61e5·39-s − 7.36e4·41-s + 4.40e5·43-s − 9.11e4·45-s − 4.65e5·47-s − 5.31e5·49-s + 6.65e5·51-s + 4.71e4·53-s + 4.48e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.595·7-s + 1/3·9-s − 0.811·11-s + 0.756·13-s + 0.258·15-s − 1.21·17-s + 1.04·19-s − 0.343·21-s − 0.0921·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 0.262·31-s + 0.468·33-s − 0.266·35-s − 0.793·37-s − 0.436·39-s − 0.166·41-s + 0.844·43-s − 0.149·45-s − 0.654·47-s − 0.645·49-s + 0.703·51-s + 0.0435·53-s + 0.363·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.345317843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345317843\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 + p^{3} T \) |
| good | 7 | \( 1 - 540 T + p^{7} T^{2} \) |
| 11 | \( 1 + 3584 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5994 T + p^{7} T^{2} \) |
| 17 | \( 1 + 24666 T + p^{7} T^{2} \) |
| 19 | \( 1 - 31276 T + p^{7} T^{2} \) |
| 23 | \( 1 + 5376 T + p^{7} T^{2} \) |
| 29 | \( 1 + 194846 T + p^{7} T^{2} \) |
| 31 | \( 1 - 43592 T + p^{7} T^{2} \) |
| 37 | \( 1 + 244358 T + p^{7} T^{2} \) |
| 41 | \( 1 + 73686 T + p^{7} T^{2} \) |
| 43 | \( 1 - 440268 T + p^{7} T^{2} \) |
| 47 | \( 1 + 465920 T + p^{7} T^{2} \) |
| 53 | \( 1 - 47154 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2289024 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1606478 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3653228 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1992832 T + p^{7} T^{2} \) |
| 73 | \( 1 + 4037070 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1942472 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1105668 T + p^{7} T^{2} \) |
| 89 | \( 1 - 14626 T + p^{7} T^{2} \) |
| 97 | \( 1 - 9367874 T + p^{7} T^{2} \) |
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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
−11.18202457564812860384904357313, −10.08490364107143473537591056192, −8.842838686148594155927639772044, −7.87331028272483794393470916764, −6.92735514885875231250617075357, −5.67028851787483316950750875288, −4.76561174657912830006573009751, −3.57647626836144364158628694052, −1.99522848704354813379823303692, −0.60928392906184059064176069645,
0.60928392906184059064176069645, 1.99522848704354813379823303692, 3.57647626836144364158628694052, 4.76561174657912830006573009751, 5.67028851787483316950750875288, 6.92735514885875231250617075357, 7.87331028272483794393470916764, 8.842838686148594155927639772044, 10.08490364107143473537591056192, 11.18202457564812860384904357313
