| L(s) = 1 | + 180·17-s − 232·19-s − 142·25-s − 108·41-s − 40·43-s − 674·49-s + 648·59-s + 232·67-s − 2.21e3·73-s − 2.30e3·83-s + 1.83e3·89-s + 380·97-s − 504·107-s + 4.42e3·113-s − 2.66e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.32e3·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 2.56·17-s − 2.80·19-s − 1.13·25-s − 0.411·41-s − 0.141·43-s − 1.96·49-s + 1.42·59-s + 0.423·67-s − 3.54·73-s − 3.04·83-s + 2.18·89-s + 0.397·97-s − 0.455·107-s + 3.68·113-s − 2·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.601·169-s + 0.000439·173-s + 0.000417·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 142 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 674 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 1322 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 116 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 13534 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 18722 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 31246 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 100106 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 51694 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 59182 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 324 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 123290 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 116 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 497666 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 1106 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 963890 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1152 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 918 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 190 T + p^{3} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.309904243605007251436130669875, −8.219196758602056358910343123426, −7.57752340317811046076004172193, −7.55954017117474441047608057536, −6.81674433646148550979848336216, −6.55701964618508951734034665874, −5.99792553621883396640395132025, −5.82292390510336463067952724686, −5.38958028929895510650081327474, −4.85234993498273774135704224725, −4.35372904206542681601505199800, −4.08240283897500379198164206280, −3.41940190106933934263112761605, −3.24860052789418178953572526777, −2.51368577855479832840806227120, −2.04294254521198085353043683477, −1.50922922084038134050162974866, −1.04645410159581640400642783804, 0, 0,
1.04645410159581640400642783804, 1.50922922084038134050162974866, 2.04294254521198085353043683477, 2.51368577855479832840806227120, 3.24860052789418178953572526777, 3.41940190106933934263112761605, 4.08240283897500379198164206280, 4.35372904206542681601505199800, 4.85234993498273774135704224725, 5.38958028929895510650081327474, 5.82292390510336463067952724686, 5.99792553621883396640395132025, 6.55701964618508951734034665874, 6.81674433646148550979848336216, 7.55954017117474441047608057536, 7.57752340317811046076004172193, 8.219196758602056358910343123426, 8.309904243605007251436130669875
