| L(s) = 1 | − 2·2-s + 12·3-s − 12·4-s − 24·6-s + 56·8-s − 54·9-s − 52·11-s − 144·12-s + 80·16-s + 452·17-s + 108·18-s + 268·19-s + 104·22-s + 672·24-s + 290·25-s − 2.05e3·27-s − 1.05e3·32-s − 624·33-s − 904·34-s + 648·36-s − 536·38-s + 1.98e3·41-s − 3.76e3·43-s + 624·44-s + 960·48-s + 962·49-s − 580·50-s + ⋯ |
| L(s) = 1 | − 1/2·2-s + 4/3·3-s − 3/4·4-s − 2/3·6-s + 7/8·8-s − 2/3·9-s − 0.429·11-s − 12-s + 5/16·16-s + 1.56·17-s + 1/3·18-s + 0.742·19-s + 0.214·22-s + 7/6·24-s + 0.463·25-s − 2.81·27-s − 1.03·32-s − 0.573·33-s − 0.782·34-s + 1/2·36-s − 0.371·38-s + 1.18·41-s − 2.03·43-s + 0.322·44-s + 5/12·48-s + 0.400·49-s − 0.231·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8674721287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8674721287\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p T + p^{4} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 p T + p^{4} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 58 p T^{2} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 962 T^{2} + p^{8} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 26 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 56162 T^{2} + p^{8} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 226 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 134 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 463682 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1298402 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 311042 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 629282 T^{2} + p^{8} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 994 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 1882 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 5320322 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1257122 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5018 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 23382242 T^{2} + p^{8} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8006 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 50512322 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 386 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 43766398 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2234 T + p^{4} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10046 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8738 T + p^{4} T^{2} )^{2} \) |
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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
−21.41974751600567597501507952751, −20.75851879792913376932067769058, −19.88017054250292570081466802057, −19.75554523375872504894024081820, −18.57549211362932464980201114502, −18.47094474270830168394216425506, −17.11849225917962360839081028424, −16.84746508040984966204166572848, −15.59975009337965161486346141718, −14.54604406493841985013454267836, −14.18644411469039162318032379072, −13.54086326319204164680456829676, −12.49505108077315281617686365600, −11.25256454262619031776370893028, −9.965584594603764605037194581979, −9.223835836731734465687979184813, −8.300859782990254159011153936074, −7.73365253042416185622777892779, −5.42938113619192888783297542980, −3.27029599977862769934910771922,
3.27029599977862769934910771922, 5.42938113619192888783297542980, 7.73365253042416185622777892779, 8.300859782990254159011153936074, 9.223835836731734465687979184813, 9.965584594603764605037194581979, 11.25256454262619031776370893028, 12.49505108077315281617686365600, 13.54086326319204164680456829676, 14.18644411469039162318032379072, 14.54604406493841985013454267836, 15.59975009337965161486346141718, 16.84746508040984966204166572848, 17.11849225917962360839081028424, 18.47094474270830168394216425506, 18.57549211362932464980201114502, 19.75554523375872504894024081820, 19.88017054250292570081466802057, 20.75851879792913376932067769058, 21.41974751600567597501507952751
