| L(s) = 1 | − 32.1·2-s − 81·3-s + 520.·4-s + 625·5-s + 2.60e3·6-s + 7.29e3·7-s − 264.·8-s + 6.56e3·9-s − 2.00e4·10-s − 7.92e4·11-s − 4.21e4·12-s + 1.48e5·13-s − 2.34e5·14-s − 5.06e4·15-s − 2.57e5·16-s − 4.03e5·17-s − 2.10e5·18-s − 1.30e5·19-s + 3.25e5·20-s − 5.90e5·21-s + 2.54e6·22-s + 1.87e6·23-s + 2.14e4·24-s + 3.90e5·25-s − 4.76e6·26-s − 5.31e5·27-s + 3.79e6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1.01·4-s + 0.447·5-s + 0.819·6-s + 1.14·7-s − 0.0228·8-s + 0.333·9-s − 0.634·10-s − 1.63·11-s − 0.586·12-s + 1.43·13-s − 1.62·14-s − 0.258·15-s − 0.983·16-s − 1.17·17-s − 0.473·18-s − 0.229·19-s + 0.454·20-s − 0.662·21-s + 2.31·22-s + 1.39·23-s + 0.0131·24-s + 0.200·25-s − 2.04·26-s − 0.192·27-s + 1.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9383866258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9383866258\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 + 32.1T + 512T^{2} \) |
| 7 | \( 1 - 7.29e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.92e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.48e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.03e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 1.87e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.55e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.02e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.13e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 8.76e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.06e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.81e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.72e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 6.20e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.26e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.09e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.12e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.93e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.63e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.23e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.62e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.40e8T + 7.60e17T^{2} \) |
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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
−10.42225837441992889778293262528, −9.196708868473622003201282487413, −8.404929579584721252218413667200, −7.68504254328550733887589756105, −6.58214406745222901100507305420, −5.39325193519507914284706887359, −4.46555903399599638431357520036, −2.48769041241462520657387353538, −1.49133106620254065270928848559, −0.59207277838887156509392874591,
0.59207277838887156509392874591, 1.49133106620254065270928848559, 2.48769041241462520657387353538, 4.46555903399599638431357520036, 5.39325193519507914284706887359, 6.58214406745222901100507305420, 7.68504254328550733887589756105, 8.404929579584721252218413667200, 9.196708868473622003201282487413, 10.42225837441992889778293262528
