| L(s) = 1 | + 4·7-s − 9-s + 2·11-s + 4·13-s + 2·17-s − 2·19-s − 25-s − 2·31-s + 2·47-s + 10·49-s − 2·53-s − 2·59-s − 2·61-s − 4·63-s − 2·67-s + 8·77-s + 81-s + 16·91-s − 2·99-s + 2·101-s − 8·113-s − 4·117-s + 8·119-s + 3·121-s + 127-s + 131-s − 8·133-s + ⋯ |
| L(s) = 1 | + 4·7-s − 9-s + 2·11-s + 4·13-s + 2·17-s − 2·19-s − 25-s − 2·31-s + 2·47-s + 10·49-s − 2·53-s − 2·59-s − 2·61-s − 4·63-s − 2·67-s + 8·77-s + 81-s + 16·91-s − 2·99-s + 2·101-s − 8·113-s − 4·117-s + 8·119-s + 3·121-s + 127-s + 131-s − 8·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.442870155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.442870155\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.23009760474550316731254113846, −6.04283894442203152032278134954, −5.92721257397977876552478806976, −5.83696953279924815092031896428, −5.80232784790860904202210208565, −5.41624867592337258551838594284, −5.17791670567486884247589183260, −4.93142220701193104095800944482, −4.87667259330048617312405283990, −4.41150643984988292977054012452, −4.22869953294293007599526234302, −4.07407524812240524836543398456, −4.02330379700013969687134688539, −3.86444550479124806060785349907, −3.48253773567159756060877599724, −3.34540680912194764454330570726, −3.05120606428891277508926800416, −2.73685768063699449893345133909, −2.26730055566263861706839171139, −2.00702952893846279146144385248, −1.66853572774740992175334328481, −1.57314495138489382055533451524, −1.44226396665154944493925426163, −1.17002816516479536543121191828, −0.996912179106803796498307249269,
0.996912179106803796498307249269, 1.17002816516479536543121191828, 1.44226396665154944493925426163, 1.57314495138489382055533451524, 1.66853572774740992175334328481, 2.00702952893846279146144385248, 2.26730055566263861706839171139, 2.73685768063699449893345133909, 3.05120606428891277508926800416, 3.34540680912194764454330570726, 3.48253773567159756060877599724, 3.86444550479124806060785349907, 4.02330379700013969687134688539, 4.07407524812240524836543398456, 4.22869953294293007599526234302, 4.41150643984988292977054012452, 4.87667259330048617312405283990, 4.93142220701193104095800944482, 5.17791670567486884247589183260, 5.41624867592337258551838594284, 5.80232784790860904202210208565, 5.83696953279924815092031896428, 5.92721257397977876552478806976, 6.04283894442203152032278134954, 6.23009760474550316731254113846
