| L(s) = 1 | − 4·13-s + 4·37-s + 4·61-s − 81-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 4·13-s + 4·37-s + 4·61-s − 81-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5743743067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5743743067\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−7.52753259997669659053556414330, −7.47448400418578472126877030440, −7.17887634069304071626328509295, −7.09799727856111460127818937676, −6.97786142857710253982331009585, −6.31510370080201837787356276565, −6.24251051548040817548455498901, −6.09232026584850991134051100787, −5.86771939584615238294315056329, −5.37356384890399184672899423801, −5.25414833489590449404043786749, −4.95346695150744839011667156222, −4.73378768843682497981537909826, −4.70122051681237414997937803449, −4.37309667551536545429944692828, −3.90513823797067780262373371644, −3.86446329139798759701981249665, −3.43294835580117660338348996374, −2.94589771446165780746346580217, −2.67488760753414508913958952736, −2.38076276436362608287248420143, −2.36187048610256578754138761460, −2.10110209145205183716226362850, −1.30094034148794529786059142175, −0.75137044797525364260919094343,
0.75137044797525364260919094343, 1.30094034148794529786059142175, 2.10110209145205183716226362850, 2.36187048610256578754138761460, 2.38076276436362608287248420143, 2.67488760753414508913958952736, 2.94589771446165780746346580217, 3.43294835580117660338348996374, 3.86446329139798759701981249665, 3.90513823797067780262373371644, 4.37309667551536545429944692828, 4.70122051681237414997937803449, 4.73378768843682497981537909826, 4.95346695150744839011667156222, 5.25414833489590449404043786749, 5.37356384890399184672899423801, 5.86771939584615238294315056329, 6.09232026584850991134051100787, 6.24251051548040817548455498901, 6.31510370080201837787356276565, 6.97786142857710253982331009585, 7.09799727856111460127818937676, 7.17887634069304071626328509295, 7.47448400418578472126877030440, 7.52753259997669659053556414330
