| L(s) = 1 | − 16·13-s + 8·25-s − 32·37-s − 32·73-s − 64·97-s + 64·109-s − 32·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + ⋯ |
| L(s) = 1 | − 4.43·13-s + 8/5·25-s − 5.26·37-s − 3.74·73-s − 6.49·97-s + 6.13·109-s − 2.90·121-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 + 3.07·169-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 + 0.0669·223-s + 0.0663·227-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06922819688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06922819688\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - 4 T^{2} + 6 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 + 16 T^{2} + 114 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 17 | \( ( 1 - 52 T^{2} + 1206 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 44 T^{2} + 1014 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 64 T^{2} + 1890 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 64 T^{2} + 2514 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 28 T^{2} + 390 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - 52 T^{2} + 3606 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 44 T^{2} + 2550 T^{4} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 71 | \( ( 1 + 160 T^{2} + 14754 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 8 T + 150 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 116 T^{2} + 10230 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 340 T^{2} + 44694 T^{4} - 340 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 + 16 T + 246 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.12023529537965489674436573428, −3.10551766956987428234742440401, −3.01938751230035147512466146343, −2.90269028603833005463153911059, −2.79397888143719388952116567187, −2.63568392018056124231595176882, −2.54961191756614407816256039069, −2.42305672388697271757029569991, −2.35019533702365575335759388101, −2.34196764905299952375797308373, −2.06526334142641229412013055848, −2.00611059052881781335732048189, −1.92377451438571335097424174388, −1.88469342789402077292551831108, −1.53917597194359588682873245510, −1.39484851824284034502269518414, −1.38905528270504097360725963186, −1.37911270995824363377505271680, −1.20244676462837898209486534882, −0.918088663860124931965254835671, −0.820278525078843737699193437896, −0.53416631046604951486466043464, −0.25780387092734817271378299086, −0.10715256345511761511550665324, −0.090879364353582949808307435677,
0.090879364353582949808307435677, 0.10715256345511761511550665324, 0.25780387092734817271378299086, 0.53416631046604951486466043464, 0.820278525078843737699193437896, 0.918088663860124931965254835671, 1.20244676462837898209486534882, 1.37911270995824363377505271680, 1.38905528270504097360725963186, 1.39484851824284034502269518414, 1.53917597194359588682873245510, 1.88469342789402077292551831108, 1.92377451438571335097424174388, 2.00611059052881781335732048189, 2.06526334142641229412013055848, 2.34196764905299952375797308373, 2.35019533702365575335759388101, 2.42305672388697271757029569991, 2.54961191756614407816256039069, 2.63568392018056124231595176882, 2.79397888143719388952116567187, 2.90269028603833005463153911059, 3.01938751230035147512466146343, 3.10551766956987428234742440401, 3.12023529537965489674436573428
Plot not available for L-functions of degree greater than 10.
