| L(s) = 1 | + 48·5-s − 238·13-s + 480·17-s + 1.67e3·25-s + 1.68e3·29-s − 2.16e3·37-s + 1.44e3·41-s + 2.40e3·49-s − 5.04e3·53-s + 6.95e3·61-s − 1.14e4·65-s − 1.44e3·73-s + 2.30e4·85-s + 1.24e4·89-s + 1.91e3·97-s − 7.92e3·101-s + 9.36e3·109-s − 6.72e3·113-s + ⋯ |
| L(s) = 1 | + 1.91·5-s − 1.40·13-s + 1.66·17-s + 2.68·25-s + 1.99·29-s − 1.57·37-s + 0.856·41-s + 49-s − 1.79·53-s + 1.86·61-s − 2.70·65-s − 0.270·73-s + 3.18·85-s + 1.57·89-s + 0.203·97-s − 0.776·101-s + 0.787·109-s − 0.526·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.223154271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.223154271\) |
| \(L(3)\) |
|
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 | \( 1 - 48 T + p^{4} T^{2} \) |
| 7 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 11 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 13 | \( 1 + 238 T + p^{4} T^{2} \) |
| 17 | \( 1 - 480 T + p^{4} T^{2} \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( 1 - 1680 T + p^{4} T^{2} \) |
| 31 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 37 | \( 1 + 2162 T + p^{4} T^{2} \) |
| 41 | \( 1 - 1440 T + p^{4} T^{2} \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 + 5040 T + p^{4} T^{2} \) |
| 59 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 61 | \( 1 - 6958 T + p^{4} T^{2} \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 1442 T + p^{4} T^{2} \) |
| 79 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 83 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 89 | \( 1 - 12480 T + p^{4} T^{2} \) |
| 97 | \( 1 - 1918 T + p^{4} T^{2} \) |
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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
−10.00368268380587850416451103018, −9.547893956759437274625076886505, −8.506686728243994076675708091019, −7.31548543608105049100002212738, −6.39816205611356354207474758417, −5.48921818711690491385773179431, −4.85137861516112031104733124721, −3.05920753706544632553908355022, −2.15895841828084299282802771540, −1.01258064246216628558872760534,
1.01258064246216628558872760534, 2.15895841828084299282802771540, 3.05920753706544632553908355022, 4.85137861516112031104733124721, 5.48921818711690491385773179431, 6.39816205611356354207474758417, 7.31548543608105049100002212738, 8.506686728243994076675708091019, 9.547893956759437274625076886505, 10.00368268380587850416451103018
