| L(s) = 1 | − 4·9-s − 50·25-s + 248·41-s + 164·49-s − 150·81-s + 568·89-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 676·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 200·225-s + ⋯ |
| L(s) = 1 | − 4/9·9-s − 2·25-s + 6.04·41-s + 3.34·49-s − 1.85·81-s + 6.38·89-s − 4·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 4·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 8/9·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.662362874\times10^{-7}\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.662362874\times10^{-7}\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2}( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 878 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1198 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{2}( 1 + 76 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4402 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 4078 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 116 T + p^{2} T^{2} )^{2}( 1 + 116 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 76 T + p^{2} T^{2} )^{2}( 1 + 76 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.65890073317884287521046556548, −6.21441538251241973290134243868, −6.19329046921102766141414708427, −6.13990248664045716787253360841, −5.88069032588630055925593379149, −5.71871526262673651444951719747, −5.35089378344739684649375227426, −5.16774888551262673665676737989, −5.10056151211076923541789858835, −4.44050091372220930313406763190, −4.40855404771172149473939239722, −4.22809411274490126313145621120, −3.95722945850504871039277967285, −3.81850526076762218552135036969, −3.45795694770782677015210768692, −3.24105441868342084118860785511, −2.75406427973830161945788341699, −2.54544096326984073703104402784, −2.31182674310274815769606450899, −2.18013348846041940590421609389, −1.93722412508759310270914217975, −1.01991485144524085662118700402, −1.00054258564060164681877071369, −0.928731717517437252013406879432, −0.000082522817669903067956342846,
0.000082522817669903067956342846, 0.928731717517437252013406879432, 1.00054258564060164681877071369, 1.01991485144524085662118700402, 1.93722412508759310270914217975, 2.18013348846041940590421609389, 2.31182674310274815769606450899, 2.54544096326984073703104402784, 2.75406427973830161945788341699, 3.24105441868342084118860785511, 3.45795694770782677015210768692, 3.81850526076762218552135036969, 3.95722945850504871039277967285, 4.22809411274490126313145621120, 4.40855404771172149473939239722, 4.44050091372220930313406763190, 5.10056151211076923541789858835, 5.16774888551262673665676737989, 5.35089378344739684649375227426, 5.71871526262673651444951719747, 5.88069032588630055925593379149, 6.13990248664045716787253360841, 6.19329046921102766141414708427, 6.21441538251241973290134243868, 6.65890073317884287521046556548
