L(s) = 1 | + 8·25-s + 16·29-s − 32·37-s + 10·49-s + 16·53-s − 18·81-s + 16·109-s + 16·113-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | + 8/5·25-s + 2.97·29-s − 5.26·37-s + 10/7·49-s + 2.19·53-s − 2·81-s + 1.53·109-s + 1.50·113-s + 3.27·121-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 + 3.07·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.085290123\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.085290123\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
good | 3 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 112 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.57780206148932177669843248804, −6.55745040968756918882947676452, −6.35957068000839944793288951178, −5.81347232657177139814617120271, −5.61375500941123443978476593547, −5.55430642434611354518589245045, −5.39676689209153270887632563842, −5.21434628590146253874550988409, −4.80363124088719428589376227549, −4.62882904224517515245740995066, −4.38150009534384657098008713447, −4.32090201298362643699747114542, −4.12277068882847408310164497820, −3.49681906302410033716260317727, −3.31979739521265682354596216654, −3.28809623886295478360444267694, −3.15454358769357933207399878380, −2.64307360221573083493627350321, −2.51231382936894167438888931944, −2.04039073266894559225728347601, −1.79214795135285516117430169771, −1.67981378046781151188319060897, −1.01050733886054927448279798545, −0.789314099347037547854179069621, −0.44586757487140784037279395617,
0.44586757487140784037279395617, 0.789314099347037547854179069621, 1.01050733886054927448279798545, 1.67981378046781151188319060897, 1.79214795135285516117430169771, 2.04039073266894559225728347601, 2.51231382936894167438888931944, 2.64307360221573083493627350321, 3.15454358769357933207399878380, 3.28809623886295478360444267694, 3.31979739521265682354596216654, 3.49681906302410033716260317727, 4.12277068882847408310164497820, 4.32090201298362643699747114542, 4.38150009534384657098008713447, 4.62882904224517515245740995066, 4.80363124088719428589376227549, 5.21434628590146253874550988409, 5.39676689209153270887632563842, 5.55430642434611354518589245045, 5.61375500941123443978476593547, 5.81347232657177139814617120271, 6.35957068000839944793288951178, 6.55745040968756918882947676452, 6.57780206148932177669843248804