| L(s) = 1 | − 4·13-s − 4·31-s + 4·43-s + 2·49-s − 4·67-s + 4·97-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 + ⋯ |
| L(s) = 1 | − 4·13-s − 4·31-s + 4·43-s + 2·49-s − 4·67-s + 4·97-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5899894966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5899894966\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
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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
−7.06455436619895528217560852122, −6.86525923532066924452135072959, −6.30731665863474286880736414604, −6.17304769351559242434115416871, −5.79240394399931854780880881073, −5.78136607915405100082449493527, −5.61786269038940756551980183896, −5.37494337854309505315929996393, −5.16747775415746864305776754321, −4.86704113590979641538307619655, −4.52027588007608767883237291715, −4.50149724821063543432988127956, −4.39978722555397598192815622328, −4.06423382406103284418400831694, −3.67198967629141611279574612688, −3.51119577129881542102389610905, −3.13588063935268142939853697068, −2.96373279465201844086410963084, −2.57643658410204539008691649513, −2.38905442266244460134343487945, −2.19137392873903803239592831973, −1.86249989552766588058839617478, −1.76854226414883281379709208327, −0.988546802046507834733748318804, −0.45533861842714877033138907550,
0.45533861842714877033138907550, 0.988546802046507834733748318804, 1.76854226414883281379709208327, 1.86249989552766588058839617478, 2.19137392873903803239592831973, 2.38905442266244460134343487945, 2.57643658410204539008691649513, 2.96373279465201844086410963084, 3.13588063935268142939853697068, 3.51119577129881542102389610905, 3.67198967629141611279574612688, 4.06423382406103284418400831694, 4.39978722555397598192815622328, 4.50149724821063543432988127956, 4.52027588007608767883237291715, 4.86704113590979641538307619655, 5.16747775415746864305776754321, 5.37494337854309505315929996393, 5.61786269038940756551980183896, 5.78136607915405100082449493527, 5.79240394399931854780880881073, 6.17304769351559242434115416871, 6.30731665863474286880736414604, 6.86525923532066924452135072959, 7.06455436619895528217560852122
