| L(s) = 1 | + 14·5-s + 208·7-s + 536·11-s − 694·13-s + 1.27e3·17-s + 1.11e3·19-s + 3.21e3·23-s − 2.92e3·25-s + 2.91e3·29-s + 2.62e3·31-s + 2.91e3·35-s + 9.45e3·37-s − 170·41-s − 1.99e4·43-s + 32·47-s + 2.64e4·49-s − 2.21e4·53-s + 7.50e3·55-s − 4.14e4·59-s − 1.54e4·61-s − 9.71e3·65-s − 2.07e4·67-s + 2.85e4·71-s − 5.36e4·73-s + 1.11e5·77-s + 6.91e4·79-s + 3.78e4·83-s + ⋯ |
| L(s) = 1 | + 0.250·5-s + 1.60·7-s + 1.33·11-s − 1.13·13-s + 1.07·17-s + 0.706·19-s + 1.26·23-s − 0.937·25-s + 0.644·29-s + 0.490·31-s + 0.401·35-s + 1.13·37-s − 0.0157·41-s − 1.64·43-s + 0.00211·47-s + 1.57·49-s − 1.08·53-s + 0.334·55-s − 1.55·59-s − 0.532·61-s − 0.285·65-s − 0.564·67-s + 0.673·71-s − 1.17·73-s + 2.14·77-s + 1.24·79-s + 0.602·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.370258611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.370258611\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 14 T + p^{5} T^{2} \) |
| 7 | \( 1 - 208 T + p^{5} T^{2} \) |
| 11 | \( 1 - 536 T + p^{5} T^{2} \) |
| 13 | \( 1 + 694 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1278 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1112 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3216 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2918 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2624 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9458 T + p^{5} T^{2} \) |
| 41 | \( 1 + 170 T + p^{5} T^{2} \) |
| 43 | \( 1 + 19928 T + p^{5} T^{2} \) |
| 47 | \( 1 - 32 T + p^{5} T^{2} \) |
| 53 | \( 1 + 22178 T + p^{5} T^{2} \) |
| 59 | \( 1 + 41480 T + p^{5} T^{2} \) |
| 61 | \( 1 + 15462 T + p^{5} T^{2} \) |
| 67 | \( 1 + 20744 T + p^{5} T^{2} \) |
| 71 | \( 1 - 28592 T + p^{5} T^{2} \) |
| 73 | \( 1 + 53670 T + p^{5} T^{2} \) |
| 79 | \( 1 - 69152 T + p^{5} T^{2} \) |
| 83 | \( 1 - 37800 T + p^{5} T^{2} \) |
| 89 | \( 1 - 126806 T + p^{5} T^{2} \) |
| 97 | \( 1 - 62290 T + p^{5} T^{2} \) |
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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
−9.835106353604234523403740006357, −9.158278785907524523590268858532, −8.041971507684710903360697080123, −7.44525012145882193863297728392, −6.30622422337821708008661614601, −5.13964257465149555562319378438, −4.54070983376368225494096612112, −3.16997111936599309308990593224, −1.79489248302274523782784334899, −0.990238674714917791665922883675,
0.990238674714917791665922883675, 1.79489248302274523782784334899, 3.16997111936599309308990593224, 4.54070983376368225494096612112, 5.13964257465149555562319378438, 6.30622422337821708008661614601, 7.44525012145882193863297728392, 8.041971507684710903360697080123, 9.158278785907524523590268858532, 9.835106353604234523403740006357
