L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 9-s + 4·13-s − 4·16-s − 2·17-s − 2·18-s − 2·25-s + 8·26-s − 2·32-s − 4·34-s − 36-s − 2·49-s − 4·50-s + 4·52-s + 2·53-s + 3·64-s − 2·68-s + 2·72-s + 81-s − 2·89-s − 4·98-s − 2·100-s + 4·101-s − 8·104-s + 4·106-s + ⋯ |
L(s) = 1 | + 2·2-s + 4-s − 2·8-s − 9-s + 4·13-s − 4·16-s − 2·17-s − 2·18-s − 2·25-s + 8·26-s − 2·32-s − 4·34-s − 36-s − 2·49-s − 4·50-s + 4·52-s + 2·53-s + 3·64-s − 2·68-s + 2·72-s + 81-s − 2·89-s − 4·98-s − 2·100-s + 4·101-s − 8·104-s + 4·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.024725000\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.024725000\) |
\(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 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
−8.295134243651476894657254720512, −8.111145523124783815237780202687, −7.79619662493473746985943868253, −7.43124846454260907082139745504, −6.91102574826430035000289349049, −6.81473999656473342397938340318, −6.47252090941402724136107794406, −6.28913316104520850601139058727, −6.00991797545547029172835080837, −5.92906387769723280787642798314, −5.91439906930685071175477783227, −5.29448475235018583592176983778, −5.27576068283942987298436104519, −4.97159460935988884613838754383, −4.43083053375474411949538090420, −4.24242304153068494756273286680, −4.01465038350638189169196351049, −3.97962228810159768289381329074, −3.39453644134763664225732626843, −3.32401147816787240906483124784, −3.19847471020486908205697602242, −2.54572849399032149803958116538, −2.17144802826572071518473476174, −1.85648996883229754103561318628, −1.08987278157440062378599310698,
1.08987278157440062378599310698, 1.85648996883229754103561318628, 2.17144802826572071518473476174, 2.54572849399032149803958116538, 3.19847471020486908205697602242, 3.32401147816787240906483124784, 3.39453644134763664225732626843, 3.97962228810159768289381329074, 4.01465038350638189169196351049, 4.24242304153068494756273286680, 4.43083053375474411949538090420, 4.97159460935988884613838754383, 5.27576068283942987298436104519, 5.29448475235018583592176983778, 5.91439906930685071175477783227, 5.92906387769723280787642798314, 6.00991797545547029172835080837, 6.28913316104520850601139058727, 6.47252090941402724136107794406, 6.81473999656473342397938340318, 6.91102574826430035000289349049, 7.43124846454260907082139745504, 7.79619662493473746985943868253, 8.111145523124783815237780202687, 8.295134243651476894657254720512