L(s) = 1 | − 8·9-s − 10·25-s − 24·29-s + 4·49-s + 30·81-s + 64·109-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 80·225-s + ⋯ |
L(s) = 1 | − 8/3·9-s − 2·25-s − 4.45·29-s + 4/7·49-s + 10/3·81-s + 6.13·109-s + 4·121-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 + 0.0783·163-s + 0.0773·167-s − 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 16/3·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4165087742\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4165087742\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
good | 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
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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
−6.18626744637382571576502939087, −6.16199721231586137252186261485, −6.15960520351229691892557538214, −5.74822505810721861857172927406, −5.55690075865132424130605400543, −5.45685524677552173695102718890, −5.40967421688039287204219122538, −4.91513420760546730349229050473, −4.75633790830764163800903661032, −4.69895265053037693066644683287, −4.06386846384179583269468146414, −4.02180998818222127083470969729, −3.81536893447775702771458449438, −3.45524386553575890633797234539, −3.43937411933734357360671393601, −3.13025275255520846070016035611, −3.03154078405596798249375947804, −2.35552498958665531634317790944, −2.25787638220547326247290733252, −2.13066588276179213650084271881, −2.05046882990868582126221356881, −1.43801247716543431557119253970, −1.15553645329425356869077312689, −0.42169270087196027488623536318, −0.18816949242081315049372663499,
0.18816949242081315049372663499, 0.42169270087196027488623536318, 1.15553645329425356869077312689, 1.43801247716543431557119253970, 2.05046882990868582126221356881, 2.13066588276179213650084271881, 2.25787638220547326247290733252, 2.35552498958665531634317790944, 3.03154078405596798249375947804, 3.13025275255520846070016035611, 3.43937411933734357360671393601, 3.45524386553575890633797234539, 3.81536893447775702771458449438, 4.02180998818222127083470969729, 4.06386846384179583269468146414, 4.69895265053037693066644683287, 4.75633790830764163800903661032, 4.91513420760546730349229050473, 5.40967421688039287204219122538, 5.45685524677552173695102718890, 5.55690075865132424130605400543, 5.74822505810721861857172927406, 6.15960520351229691892557538214, 6.16199721231586137252186261485, 6.18626744637382571576502939087