| L(s) = 1 | + 11.3·2-s + 128.·4-s + 1.03e3i·5-s + (−283. − 2.38e3i)7-s + 1.44e3·8-s + 1.16e4i·10-s − 2.45e4·11-s − 3.16e4i·13-s + (−3.21e3 − 2.69e4i)14-s + 1.63e4·16-s − 7.10e4i·17-s − 5.13e4i·19-s + 1.31e5i·20-s − 2.77e5·22-s + 1.04e5·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + 1.64i·5-s + (−0.118 − 0.992i)7-s + 0.353·8-s + 1.16i·10-s − 1.67·11-s − 1.10i·13-s + (−0.0836 − 0.702i)14-s + 0.250·16-s − 0.851i·17-s − 0.394i·19-s + 0.824i·20-s − 1.18·22-s + 0.374·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.118 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.906611461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.906611461\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 11.3T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (283. + 2.38e3i)T \) |
| good | 5 | \( 1 - 1.03e3iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 2.45e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 3.16e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 7.10e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 5.13e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.04e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 1.08e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + 5.73e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.41e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.61e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 6.14e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + 9.15e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.19e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.03e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 8.60e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 5.17e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.02e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 3.52e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 4.76e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 3.60e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 9.28e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.17e8iT - 7.83e15T^{2} \) |
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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
−11.36574677653574377842382778221, −10.58722939655032176380097848940, −10.08392020444422221066501983206, −7.80828350839830646656872171773, −7.22425738513785873867506605338, −6.08144887154706464520449375144, −4.75723200999714198116621618036, −3.17610617173330936975425847474, −2.63585811148432292010221981706, −0.37763373962486709716993182570,
1.39210037879933497233159104866, 2.67367047238196665378085899701, 4.44585184854578687753208661284, 5.19084921492672752115454784371, 6.19090098932464086444498596070, 7.978347442508855681369554000430, 8.735950425264344849852239007702, 9.917843721853863225552305615409, 11.34820206109498514818868741932, 12.48953712211389129557607471196
