| L(s) = 1 | + 9-s − 4·17-s + 2·25-s − 2·41-s − 2·49-s + 4·73-s − 8·89-s + 2·97-s − 4·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 9-s − 4·17-s + 2·25-s − 2·41-s − 2·49-s + 4·73-s − 8·89-s + 2·97-s − 4·113-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7344661874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7344661874\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_1$ | \( ( 1 + T )^{8} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−6.72446234221896453256723010868, −6.32502592449527237233284336875, −6.31785400950188089241628900101, −6.30507226751181407969253879913, −5.74322329109195418353263847857, −5.43584338805768732541667406257, −5.15308494852303517004346873195, −5.06328520266366006865304038487, −5.02403469423489059128847883666, −4.79685073660057679264865192646, −4.29334579426861896324411497088, −4.29319001671565954305860173760, −4.07922018546408368656316682214, −3.98250763144102804436512736831, −3.71250563335131941767922600289, −3.22886789385100624913097128544, −2.91058592626023877208724248012, −2.90021867262857203280278438699, −2.57929052012426391999492124663, −2.31656697144977186146516788183, −1.95744892942415873981569582460, −1.66138205462997518508901802122, −1.53600053290924852594575501925, −1.15721029058040105332646797819, −0.39066463209831322747311860897,
0.39066463209831322747311860897, 1.15721029058040105332646797819, 1.53600053290924852594575501925, 1.66138205462997518508901802122, 1.95744892942415873981569582460, 2.31656697144977186146516788183, 2.57929052012426391999492124663, 2.90021867262857203280278438699, 2.91058592626023877208724248012, 3.22886789385100624913097128544, 3.71250563335131941767922600289, 3.98250763144102804436512736831, 4.07922018546408368656316682214, 4.29319001671565954305860173760, 4.29334579426861896324411497088, 4.79685073660057679264865192646, 5.02403469423489059128847883666, 5.06328520266366006865304038487, 5.15308494852303517004346873195, 5.43584338805768732541667406257, 5.74322329109195418353263847857, 6.30507226751181407969253879913, 6.31785400950188089241628900101, 6.32502592449527237233284336875, 6.72446234221896453256723010868
