L(s) = 1 | − 16·17-s + 32·41-s − 16·49-s + 16·73-s − 18·81-s − 8·89-s + 16·97-s + 64·113-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | − 3.88·17-s + 4.99·41-s − 2.28·49-s + 1.87·73-s − 2·81-s − 0.847·89-s + 1.62·97-s + 6.02·113-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.535056141\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.535056141\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.08797486155196423425015904944, −5.99068701587077288274577183458, −5.87634819247458266645791226122, −5.67327208512409853972541105525, −5.30771855816824130925688569172, −4.93037169443121535536541115174, −4.78763567242103038988951752184, −4.70657617482620261566458637429, −4.64129173711993029500116854503, −4.23666028465203257760912094100, −4.07341132925047361482517083902, −3.94470978003321719287294247891, −3.87649226425293523423832340623, −3.25919946586406042264397123034, −3.18162717407602784298173151095, −2.87261124763231417361232692252, −2.63590278244596198475322251602, −2.49419772232160104124188838813, −2.06079137230374219401437132078, −2.05052485174579993904240380620, −1.73982622126437590290783319049, −1.50429610665218182842673689060, −0.74456486605857232212256740680, −0.64787843041101306061059246520, −0.39458595484062388352807966595,
0.39458595484062388352807966595, 0.64787843041101306061059246520, 0.74456486605857232212256740680, 1.50429610665218182842673689060, 1.73982622126437590290783319049, 2.05052485174579993904240380620, 2.06079137230374219401437132078, 2.49419772232160104124188838813, 2.63590278244596198475322251602, 2.87261124763231417361232692252, 3.18162717407602784298173151095, 3.25919946586406042264397123034, 3.87649226425293523423832340623, 3.94470978003321719287294247891, 4.07341132925047361482517083902, 4.23666028465203257760912094100, 4.64129173711993029500116854503, 4.70657617482620261566458637429, 4.78763567242103038988951752184, 4.93037169443121535536541115174, 5.30771855816824130925688569172, 5.67327208512409853972541105525, 5.87634819247458266645791226122, 5.99068701587077288274577183458, 6.08797486155196423425015904944