| L(s) = 1 | − 2-s − 2·4-s − 4·7-s + 3·8-s + 6·11-s − 2·13-s + 4·14-s + 16-s + 4·17-s − 6·22-s − 13·23-s + 2·26-s + 8·28-s + 5·29-s + 4·31-s − 2·32-s − 4·34-s − 4·37-s + 21·41-s − 17·43-s − 12·44-s + 13·46-s + 14·47-s − 2·49-s + 4·52-s − 18·53-s − 12·56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s − 1.51·7-s + 1.06·8-s + 1.80·11-s − 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.970·17-s − 1.27·22-s − 2.71·23-s + 0.392·26-s + 1.51·28-s + 0.928·29-s + 0.718·31-s − 0.353·32-s − 0.685·34-s − 0.657·37-s + 3.27·41-s − 2.59·43-s − 1.80·44-s + 1.91·46-s + 2.04·47-s − 2/7·49-s + 0.554·52-s − 2.47·53-s − 1.60·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 33 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 13 T + 87 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_4$ | \( 1 - 21 T + 191 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 17 T + 157 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 138 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 18 T + 182 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 63 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 81 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 9 T + 131 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 25 T + 303 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 183 T^{2} - p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.058761387913769390759206754715, −7.75574510743999089354799921660, −7.18944872473669821507225983523, −7.02153086636121786566493303414, −6.45184662222692698774381120346, −6.16140025949265467011638416969, −5.97871628363432092026064896772, −5.64670458012498577193518659701, −4.84624068738795639397808224097, −4.68400757169133059369707768368, −4.14178004347461465395548155382, −3.93205671171735489237269264874, −3.54728059228494985462628079567, −3.14195708488296106043979789671, −2.56768649451905884485716367257, −2.08136400647833211258189485686, −1.22155713237536775014526789573, −1.09820080009033362205797219087, 0, 0,
1.09820080009033362205797219087, 1.22155713237536775014526789573, 2.08136400647833211258189485686, 2.56768649451905884485716367257, 3.14195708488296106043979789671, 3.54728059228494985462628079567, 3.93205671171735489237269264874, 4.14178004347461465395548155382, 4.68400757169133059369707768368, 4.84624068738795639397808224097, 5.64670458012498577193518659701, 5.97871628363432092026064896772, 6.16140025949265467011638416969, 6.45184662222692698774381120346, 7.02153086636121786566493303414, 7.18944872473669821507225983523, 7.75574510743999089354799921660, 8.058761387913769390759206754715
