| L(s) = 1 | + 4·7-s − 8·17-s + 4·23-s + 6·25-s + 8·31-s + 16·41-s + 10·49-s − 32·71-s − 24·73-s + 32·89-s + 16·97-s + 24·103-s − 36·113-s − 32·119-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 16·161-s + 163-s + 167-s + 38·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.94·17-s + 0.834·23-s + 6/5·25-s + 1.43·31-s + 2.49·41-s + 10/7·49-s − 3.79·71-s − 2.80·73-s + 3.39·89-s + 1.62·97-s + 2.36·103-s − 3.38·113-s − 2.93·119-s + 2.18·121-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 + 1.26·161-s + 0.0783·163-s + 0.0773·167-s + 2.92·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.255886484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.255886484\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 6 T^{2} + 42 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 24 T^{2} + 318 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 38 T^{2} + 682 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 66 T^{2} + 1794 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 76 T^{2} + 2854 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 108 T^{2} + 5382 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 152 T^{2} + 9406 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 - 132 T^{2} + 8886 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 178 T^{2} + 14050 T^{4} - 178 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 230 T^{2} + 20650 T^{4} - 230 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 112 T^{2} + 8782 T^{4} + 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 210 T^{2} + 23970 T^{4} - 210 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.51764247666005581458441036784, −6.38147914063714069519996233491, −6.14833206038043373428517769538, −5.76588317463883117822290661060, −5.62069039025514517377220295506, −5.59866198884309752821810617004, −5.13090608284422109886483636766, −4.87993866831912458212123275877, −4.80318761115358609887035101031, −4.49505969742057791271537818204, −4.42741692198653230391357747498, −4.32546040138722898843961976623, −3.98726696512071061605096287370, −3.79525392278997373085818287654, −3.25011517452681407833633006328, −3.10343184899661398254866336670, −2.85986157711299465254016470925, −2.57176436146579307725274319195, −2.50088796639934859967725263138, −1.94721064272530246330563546755, −1.88705717520295758264284632025, −1.46762211652819017052670687492, −1.17144276356073389244161082590, −0.74570935609150288092788617692, −0.45469581907457320733457108162,
0.45469581907457320733457108162, 0.74570935609150288092788617692, 1.17144276356073389244161082590, 1.46762211652819017052670687492, 1.88705717520295758264284632025, 1.94721064272530246330563546755, 2.50088796639934859967725263138, 2.57176436146579307725274319195, 2.85986157711299465254016470925, 3.10343184899661398254866336670, 3.25011517452681407833633006328, 3.79525392278997373085818287654, 3.98726696512071061605096287370, 4.32546040138722898843961976623, 4.42741692198653230391357747498, 4.49505969742057791271537818204, 4.80318761115358609887035101031, 4.87993866831912458212123275877, 5.13090608284422109886483636766, 5.59866198884309752821810617004, 5.62069039025514517377220295506, 5.76588317463883117822290661060, 6.14833206038043373428517769538, 6.38147914063714069519996233491, 6.51764247666005581458441036784
