| L(s) = 1 | − 2·2-s + 4-s + 2·8-s + 9-s − 2·11-s − 4·16-s − 2·18-s + 4·22-s − 2·25-s + 2·32-s + 36-s − 2·43-s − 2·44-s + 4·50-s + 3·64-s + 2·67-s + 2·72-s + 4·86-s − 4·88-s − 2·99-s − 2·100-s − 4·107-s + 4·113-s + 3·121-s + 127-s − 6·128-s + 131-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 2·8-s + 9-s − 2·11-s − 4·16-s − 2·18-s + 4·22-s − 2·25-s + 2·32-s + 36-s − 2·43-s − 2·44-s + 4·50-s + 3·64-s + 2·67-s + 2·72-s + 4·86-s − 4·88-s − 2·99-s − 2·100-s − 4·107-s + 4·113-s + 3·121-s + 127-s − 6·128-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2058465385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2058465385\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.20208793300539520749907447819, −6.12290762513905913767447961809, −6.06612146436700237984139863594, −5.49412646113211032712963826944, −5.34965341663820322603749321558, −5.33669425285980143677283930936, −5.05920633259048110981569120012, −4.82505034118184843350544657317, −4.73035543634894540203255999174, −4.50862723951997519328561710409, −4.25714491406107123714254798086, −3.91005677552811266864581172580, −3.71835541975936937054137983387, −3.70138241961071800714955637713, −3.50462363960872984512341434575, −2.94539513307650878063401173345, −2.68603060676965850021642576473, −2.56839747743186034436499420823, −2.22979918372010127398349688897, −1.99808935674086793156071730533, −1.64938834734945409537112654680, −1.56910118947029171975943124022, −1.27237444306253006333355946687, −0.70994332287186614771724499622, −0.34206758284333131160086962497,
0.34206758284333131160086962497, 0.70994332287186614771724499622, 1.27237444306253006333355946687, 1.56910118947029171975943124022, 1.64938834734945409537112654680, 1.99808935674086793156071730533, 2.22979918372010127398349688897, 2.56839747743186034436499420823, 2.68603060676965850021642576473, 2.94539513307650878063401173345, 3.50462363960872984512341434575, 3.70138241961071800714955637713, 3.71835541975936937054137983387, 3.91005677552811266864581172580, 4.25714491406107123714254798086, 4.50862723951997519328561710409, 4.73035543634894540203255999174, 4.82505034118184843350544657317, 5.05920633259048110981569120012, 5.33669425285980143677283930936, 5.34965341663820322603749321558, 5.49412646113211032712963826944, 6.06612146436700237984139863594, 6.12290762513905913767447961809, 6.20208793300539520749907447819
