| L(s) = 1 | − 4·3-s + 12·7-s + 8·9-s − 12·13-s − 48·21-s + 8·25-s − 12·27-s − 16·31-s + 16·37-s + 48·39-s + 20·43-s + 72·49-s + 8·61-s + 96·63-s + 28·67-s + 4·73-s − 32·75-s + 23·81-s − 144·91-s + 64·93-s + 48·97-s − 52·103-s − 64·111-s − 96·117-s + 28·121-s + 127-s − 80·129-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 4.53·7-s + 8/3·9-s − 3.32·13-s − 10.4·21-s + 8/5·25-s − 2.30·27-s − 2.87·31-s + 2.63·37-s + 7.68·39-s + 3.04·43-s + 72/7·49-s + 1.02·61-s + 12.0·63-s + 3.42·67-s + 0.468·73-s − 3.69·75-s + 23/9·81-s − 15.0·91-s + 6.63·93-s + 4.87·97-s − 5.12·103-s − 6.07·111-s − 8.87·117-s + 2.54·121-s + 0.0887·127-s − 7.04·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.870523457\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.870523457\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 3518 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 100 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.92698186394244198211899544682, −6.47019542929063949458821509815, −6.39903365804757882104017711584, −6.17283665100780880612427368737, −5.81927688663643617064548646352, −5.46217985153158451086782595911, −5.35626394802433800821904376365, −5.25089025812466061772392331390, −5.00082338016882856162623672124, −4.99802435789963852958869090990, −4.94284588916573329290900683996, −4.50571862223244721734267275288, −4.24297701650165261364834819024, −4.13476556221293429666347153850, −3.97141231131997316245220540232, −3.58002301685621773609168675374, −2.78658373639082937498506049695, −2.54746560589823977563982490788, −2.54535488156820311773051452545, −2.00357536431767886759296281776, −1.85705099346706542136574653442, −1.69486811636955110913634629832, −1.04513868396122868777875087781, −0.74571688841136851292583895695, −0.55382754662239501662072083754,
0.55382754662239501662072083754, 0.74571688841136851292583895695, 1.04513868396122868777875087781, 1.69486811636955110913634629832, 1.85705099346706542136574653442, 2.00357536431767886759296281776, 2.54535488156820311773051452545, 2.54746560589823977563982490788, 2.78658373639082937498506049695, 3.58002301685621773609168675374, 3.97141231131997316245220540232, 4.13476556221293429666347153850, 4.24297701650165261364834819024, 4.50571862223244721734267275288, 4.94284588916573329290900683996, 4.99802435789963852958869090990, 5.00082338016882856162623672124, 5.25089025812466061772392331390, 5.35626394802433800821904376365, 5.46217985153158451086782595911, 5.81927688663643617064548646352, 6.17283665100780880612427368737, 6.39903365804757882104017711584, 6.47019542929063949458821509815, 6.92698186394244198211899544682
