| L(s) = 1 | + 4·2-s + 4·4-s − 16·8-s + 4·11-s − 64·16-s + 16·22-s + 60·23-s + 18·25-s − 848·29-s − 64·32-s − 492·37-s − 1.13e3·43-s + 16·44-s + 240·46-s + 72·50-s + 1.09e3·53-s − 3.39e3·58-s + 192·64-s − 1.30e3·67-s − 3.08e3·71-s − 1.96e3·74-s − 944·79-s − 4.54e3·86-s − 64·88-s + 240·92-s + 72·100-s + 4.38e3·106-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.707·8-s + 0.109·11-s − 16-s + 0.155·22-s + 0.543·23-s + 0.143·25-s − 5.42·29-s − 0.353·32-s − 2.18·37-s − 4.02·43-s + 0.0548·44-s + 0.769·46-s + 0.203·50-s + 2.84·53-s − 7.67·58-s + 3/8·64-s − 2.37·67-s − 5.14·71-s − 3.09·74-s − 1.34·79-s − 5.69·86-s − 0.0775·88-s + 0.271·92-s + 0.0719·100-s + 4.01·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1782226333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1782226333\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^3$ | \( 1 - 18 T^{2} - 15301 T^{4} - 18 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 1327 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 3466 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 7738 T^{2} + 35739075 T^{4} - 7738 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 9482 T^{2} + 42862443 T^{4} + 9482 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T - 11267 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 212 T + p^{3} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 14110 T^{2} - 688411581 T^{4} - 14110 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 246 T + 9863 T^{2} + 246 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 35530 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 284 T + p^{3} T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 203934 T^{2} + 30809861027 T^{4} - 203934 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 548 T + 151427 T^{2} - 548 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 38394 T^{2} - 40706434405 T^{4} + 38394 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 185770 T^{2} - 17009881461 T^{4} - 185770 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 652 T + 124341 T^{2} + 652 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 770 T + p^{3} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 172238 T^{2} - 121668297645 T^{4} + 172238 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 472 T - 270255 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 1110166 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 897450 T^{2} + 308435211539 T^{4} - 897450 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 1732546 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−6.94630566046592108623494841632, −6.68407715637186431101662847553, −6.25346904542678486950426535266, −6.09727640575604641066788536107, −5.73963712269454542812193534845, −5.56537778095976981885457314198, −5.46693889708408026343265023005, −5.24103933116025458849747377254, −5.21584634831465477775720968772, −4.69056241883356297286603984709, −4.47634909195130006488763853507, −4.11718552889467539479514561367, −4.08872921556467143489206129006, −3.76579400741589641895325400437, −3.42835621546613244156668145739, −3.39341565055083661245140394450, −2.96446988163331322804508191210, −2.87475266732573654092380543068, −2.29952039664048199560811616951, −2.05809951818547145869280592465, −1.53336759996161026529063834937, −1.47819903659126465333033178469, −1.36432109454915316902918941861, −0.21565718342853067837601354574, −0.097943365115504219589888462407,
0.097943365115504219589888462407, 0.21565718342853067837601354574, 1.36432109454915316902918941861, 1.47819903659126465333033178469, 1.53336759996161026529063834937, 2.05809951818547145869280592465, 2.29952039664048199560811616951, 2.87475266732573654092380543068, 2.96446988163331322804508191210, 3.39341565055083661245140394450, 3.42835621546613244156668145739, 3.76579400741589641895325400437, 4.08872921556467143489206129006, 4.11718552889467539479514561367, 4.47634909195130006488763853507, 4.69056241883356297286603984709, 5.21584634831465477775720968772, 5.24103933116025458849747377254, 5.46693889708408026343265023005, 5.56537778095976981885457314198, 5.73963712269454542812193534845, 6.09727640575604641066788536107, 6.25346904542678486950426535266, 6.68407715637186431101662847553, 6.94630566046592108623494841632
