| L(s) = 1 | + 6·3-s + 4·5-s + 20·7-s + 27·9-s + 20·11-s + 26·13-s + 24·15-s − 68·17-s − 28·19-s + 120·21-s + 120·23-s − 18·25-s + 108·27-s + 12·29-s + 460·31-s + 120·33-s + 80·35-s − 228·37-s + 156·39-s + 92·41-s − 432·43-s + 108·45-s + 716·47-s − 166·49-s − 408·51-s + 356·53-s + 80·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.357·5-s + 1.07·7-s + 9-s + 0.548·11-s + 0.554·13-s + 0.413·15-s − 0.970·17-s − 0.338·19-s + 1.24·21-s + 1.08·23-s − 0.143·25-s + 0.769·27-s + 0.0768·29-s + 2.66·31-s + 0.633·33-s + 0.386·35-s − 1.01·37-s + 0.640·39-s + 0.350·41-s − 1.53·43-s + 0.357·45-s + 2.22·47-s − 0.483·49-s − 1.12·51-s + 0.922·53-s + 0.196·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.12339117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.12339117\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 4 T + 34 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 20 T + 566 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 20 T + 1882 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 p T + 10102 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 28 T + 13694 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 120 T + 24414 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 460 T + 110502 T^{2} - 460 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 228 T + 113422 T^{2} + 228 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 92 T + 113338 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 432 T + 183670 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 716 T + 321730 T^{2} - 716 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 356 T + 273118 T^{2} - 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 78 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 204 T - 42514 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 204 T + 39710 T^{2} + 204 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 604 T + 799106 T^{2} + 604 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 484 T + 730118 T^{2} - 484 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1760 T + 1672478 T^{2} - 1760 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 268 T + 1073530 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 76 T - 491398 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 1183830 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.822875750670424011812108817378, −8.580246664667339829285116039000, −7.938524458408277129830673652689, −7.78734772325405880652713571009, −7.40208562310227222594430794238, −6.73966352018207917098514398002, −6.42035717364235713458222382026, −6.41541411065145478510039068912, −5.56862081152211823733458616465, −5.11573464981310914896124032192, −4.58776790855531601925819629576, −4.58613120359733718857407543220, −3.79069705725521768611067178115, −3.63984358168551239187237838088, −2.81809388124341308765377314636, −2.65025756616998743396575951158, −1.84493762160288797822191952282, −1.81569171989624458601438818509, −0.964535231375777525846629152894, −0.67560082620932839629182886096,
0.67560082620932839629182886096, 0.964535231375777525846629152894, 1.81569171989624458601438818509, 1.84493762160288797822191952282, 2.65025756616998743396575951158, 2.81809388124341308765377314636, 3.63984358168551239187237838088, 3.79069705725521768611067178115, 4.58613120359733718857407543220, 4.58776790855531601925819629576, 5.11573464981310914896124032192, 5.56862081152211823733458616465, 6.41541411065145478510039068912, 6.42035717364235713458222382026, 6.73966352018207917098514398002, 7.40208562310227222594430794238, 7.78734772325405880652713571009, 7.938524458408277129830673652689, 8.580246664667339829285116039000, 8.822875750670424011812108817378
