| L(s) = 1 | + 4·2-s + 8·4-s + 4·5-s + 8·8-s + 16·10-s + 8·13-s − 4·16-s − 8·17-s + 32·20-s + 2·25-s + 32·26-s − 32·32-s − 32·34-s + 32·40-s − 12·41-s + 8·50-s + 64·52-s − 20·53-s + 12·61-s − 64·64-s + 32·65-s − 64·68-s − 8·73-s − 16·80-s + 17·81-s − 48·82-s − 32·85-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 1.78·5-s + 2.82·8-s + 5.05·10-s + 2.21·13-s − 16-s − 1.94·17-s + 7.15·20-s + 2/5·25-s + 6.27·26-s − 5.65·32-s − 5.48·34-s + 5.05·40-s − 1.87·41-s + 1.13·50-s + 8.87·52-s − 2.74·53-s + 1.53·61-s − 8·64-s + 3.96·65-s − 7.76·68-s − 0.936·73-s − 1.78·80-s + 17/9·81-s − 5.30·82-s − 3.47·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.33792507\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.33792507\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^3$ | \( 1 - 17 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 463 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 3169 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 4094 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 104 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 7297 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 144 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 11503 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 169 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.96661745163708795440583247417, −6.71265845972491309248403239695, −6.38896508181811347823610378470, −6.30663809413017488527383008206, −6.29627974462181733893250877655, −5.93544081267595898533649230330, −5.90063986381706458399704365638, −5.41177410184976943294425704182, −5.20049193047179363259453585248, −5.18355142087250553172659079946, −5.08039987569420161313936121845, −4.48834611034718732138924755206, −4.29008820603154494556062455561, −4.22436837931671521862636270758, −3.88187010275120483006346747648, −3.73592158438297313713361948749, −3.29548155460522673174264281235, −3.02014210054824986235916359744, −3.00486959749913477950328780813, −2.57023814195541647673460153247, −1.99098410175885145431026118226, −1.93616226989996280484135079500, −1.84223054606729412475250656292, −1.31301244491901548217199230604, −0.43443553181865036466808299904,
0.43443553181865036466808299904, 1.31301244491901548217199230604, 1.84223054606729412475250656292, 1.93616226989996280484135079500, 1.99098410175885145431026118226, 2.57023814195541647673460153247, 3.00486959749913477950328780813, 3.02014210054824986235916359744, 3.29548155460522673174264281235, 3.73592158438297313713361948749, 3.88187010275120483006346747648, 4.22436837931671521862636270758, 4.29008820603154494556062455561, 4.48834611034718732138924755206, 5.08039987569420161313936121845, 5.18355142087250553172659079946, 5.20049193047179363259453585248, 5.41177410184976943294425704182, 5.90063986381706458399704365638, 5.93544081267595898533649230330, 6.29627974462181733893250877655, 6.30663809413017488527383008206, 6.38896508181811347823610378470, 6.71265845972491309248403239695, 6.96661745163708795440583247417
