| L(s) = 1 | + 4·3-s + 2·4-s − 8·7-s + 6·9-s + 8·12-s − 8·13-s + 16·19-s − 32·21-s + 16·25-s − 4·27-s − 16·28-s + 12·36-s − 32·39-s + 34·49-s − 16·52-s + 64·57-s + 40·61-s − 48·63-s − 8·64-s + 64·75-s + 32·76-s + 32·79-s − 37·81-s − 64·84-s + 64·91-s + 32·100-s − 8·108-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 4-s − 3.02·7-s + 2·9-s + 2.30·12-s − 2.21·13-s + 3.67·19-s − 6.98·21-s + 16/5·25-s − 0.769·27-s − 3.02·28-s + 2·36-s − 5.12·39-s + 34/7·49-s − 2.21·52-s + 8.47·57-s + 5.12·61-s − 6.04·63-s − 64-s + 7.39·75-s + 3.67·76-s + 3.60·79-s − 4.11·81-s − 6.98·84-s + 6.70·91-s + 16/5·100-s − 0.769·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.543274576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.543274576\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.228128192726993745966332214796, −9.176654591269148900012949611976, −9.163532332900421735715782390263, −8.731822711244444371767968159318, −8.205088192316876085636137458578, −8.110675502425022742908573129188, −7.61488125754655462796501116452, −7.30751051246633701970792744583, −7.29874322824967643759055039025, −7.08140929354083034200814023059, −6.56605561701785345537505526240, −6.53683517156225562797071974637, −6.30682349472081109387819837861, −5.37419154995384350458999495620, −5.32668073147634338698634020785, −5.26794249345049940889028434929, −4.68324490104985024585615052636, −3.75897831318437310597711497343, −3.71014308584302199155787165860, −3.39157678668300922816589119719, −3.01008435776676640178231297580, −2.67273148817510130623183120037, −2.57929940716044093213498160435, −2.33713640408367498360285531025, −1.00485030685704020045278654916,
1.00485030685704020045278654916, 2.33713640408367498360285531025, 2.57929940716044093213498160435, 2.67273148817510130623183120037, 3.01008435776676640178231297580, 3.39157678668300922816589119719, 3.71014308584302199155787165860, 3.75897831318437310597711497343, 4.68324490104985024585615052636, 5.26794249345049940889028434929, 5.32668073147634338698634020785, 5.37419154995384350458999495620, 6.30682349472081109387819837861, 6.53683517156225562797071974637, 6.56605561701785345537505526240, 7.08140929354083034200814023059, 7.29874322824967643759055039025, 7.30751051246633701970792744583, 7.61488125754655462796501116452, 8.110675502425022742908573129188, 8.205088192316876085636137458578, 8.731822711244444371767968159318, 9.163532332900421735715782390263, 9.176654591269148900012949611976, 9.228128192726993745966332214796
