| L(s) = 1 | − 4·4-s + 12·16-s − 4·19-s − 20·25-s + 20·43-s + 28·49-s − 32·64-s − 28·67-s − 4·73-s + 16·76-s + 20·97-s + 80·100-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s − 80·172-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s − 0.917·19-s − 4·25-s + 3.04·43-s + 4·49-s − 4·64-s − 3.42·67-s − 0.468·73-s + 1.83·76-s + 2.03·97-s + 8·100-s − 1.27·121-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 − 6.09·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9940479530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9940479530\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.74291470598763338407748441240, −7.46643886160990206359066577902, −7.25739446555432652480300084844, −7.10548112435564205428107920709, −6.47900419701703289200527048364, −6.40152793338085444395816460526, −5.82729715202728282075681963376, −5.82310435506547778896802177956, −5.80561055127017851524459833417, −5.52994391536927278373923929330, −5.34432275054766543978739586691, −4.75398982015389886869760070489, −4.51615354657652496836115593356, −4.33527821899086654859756035047, −4.07879094481029168473165058692, −3.95205127966391361712074758796, −3.89298545530081624117668937954, −3.25221440944269677235185566508, −3.08629931544099355842589160802, −2.61952994936141441390227719773, −2.18237577280303330503605838984, −1.92210431985025915873400622476, −1.48640331000378513338915940661, −0.72887348860517247770646005441, −0.43824990547743013496252297095,
0.43824990547743013496252297095, 0.72887348860517247770646005441, 1.48640331000378513338915940661, 1.92210431985025915873400622476, 2.18237577280303330503605838984, 2.61952994936141441390227719773, 3.08629931544099355842589160802, 3.25221440944269677235185566508, 3.89298545530081624117668937954, 3.95205127966391361712074758796, 4.07879094481029168473165058692, 4.33527821899086654859756035047, 4.51615354657652496836115593356, 4.75398982015389886869760070489, 5.34432275054766543978739586691, 5.52994391536927278373923929330, 5.80561055127017851524459833417, 5.82310435506547778896802177956, 5.82729715202728282075681963376, 6.40152793338085444395816460526, 6.47900419701703289200527048364, 7.10548112435564205428107920709, 7.25739446555432652480300084844, 7.46643886160990206359066577902, 7.74291470598763338407748441240
