| L(s) = 1 | + 2·5-s − 7-s − 2·11-s − 4·13-s − 4·17-s − 19-s − 25-s − 2·29-s + 10·31-s − 2·35-s + 4·37-s + 6·41-s − 12·43-s + 8·47-s + 49-s − 6·53-s − 4·55-s − 2·61-s − 8·65-s + 4·67-s + 14·73-s + 2·77-s − 2·79-s − 6·83-s − 8·85-s + 14·89-s + 4·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 0.603·11-s − 1.10·13-s − 0.970·17-s − 0.229·19-s − 1/5·25-s − 0.371·29-s + 1.79·31-s − 0.338·35-s + 0.657·37-s + 0.937·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.539·55-s − 0.256·61-s − 0.992·65-s + 0.488·67-s + 1.63·73-s + 0.227·77-s − 0.225·79-s − 0.658·83-s − 0.867·85-s + 1.48·89-s + 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.689862518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.689862518\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.68557191994547128765200566707, −6.90799886071575082770814903697, −6.31079891171523755375173823430, −5.72575598927508702257809681051, −4.88346535395088514858425215423, −4.41122731915507956078540260279, −3.26758845643958718954852683396, −2.45071305892820956582600252376, −1.99030459268805120222651208830, −0.58953364281538850915737457809,
0.58953364281538850915737457809, 1.99030459268805120222651208830, 2.45071305892820956582600252376, 3.26758845643958718954852683396, 4.41122731915507956078540260279, 4.88346535395088514858425215423, 5.72575598927508702257809681051, 6.31079891171523755375173823430, 6.90799886071575082770814903697, 7.68557191994547128765200566707
