| L(s) = 1 | − 3·3-s − 7·5-s + 35·7-s + 7·11-s − 104·13-s + 21·15-s − 72·17-s + 20·19-s − 105·21-s − 48·23-s + 125·25-s + 27·27-s − 486·29-s + 95·31-s − 21·33-s − 245·35-s − 352·37-s + 312·39-s − 592·41-s − 316·43-s − 142·47-s + 882·49-s + 216·51-s + 375·53-s − 49·55-s − 60·57-s + 279·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.626·5-s + 1.88·7-s + 0.191·11-s − 2.21·13-s + 0.361·15-s − 1.02·17-s + 0.241·19-s − 1.09·21-s − 0.435·23-s + 25-s + 0.192·27-s − 3.11·29-s + 0.550·31-s − 0.110·33-s − 1.18·35-s − 1.56·37-s + 1.28·39-s − 2.25·41-s − 1.12·43-s − 0.440·47-s + 18/7·49-s + 0.593·51-s + 0.971·53-s − 0.120·55-s − 0.139·57-s + 0.615·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3508701812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3508701812\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 - 5 p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T - 76 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 7 T - 1282 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 72 T + 271 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 20 T - 6459 T^{2} - 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 48 T - 9863 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 243 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 95 T - 20766 T^{2} - 95 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 352 T + 73251 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 296 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 158 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 142 T - 83659 T^{2} + 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 375 T - 8252 T^{2} - 375 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 279 T - 127538 T^{2} - 279 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 246 T - 166465 T^{2} + 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 730 T + 232137 T^{2} + 730 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 338 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 542 T - 95253 T^{2} - 542 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 305 T - 400014 T^{2} + 305 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1123 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 426 T - 523493 T^{2} - 426 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 369 T + p^{3} T^{2} )^{2} \) |
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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
−11.61539604215763249739254180093, −10.79390716009387905853506156502, −10.77236707031257093073762291445, −9.971478098983298193625478616046, −9.635399519932249496501648038808, −8.836370075062313285927782482062, −8.542942690729029615989297886583, −8.061353232996184694510439069890, −7.44032182163567959265361623952, −6.99126396212536506782715607536, −6.95088234118917892805002856297, −5.58562915736206628470840563453, −5.48923003553805115382807850939, −4.69685100634529141342710886258, −4.64102308332650932047665131636, −3.82742139000405564426332305807, −2.95793528559440532826766385016, −1.94189992670889740080811254251, −1.68756826127567366277098844379, −0.21074069435558627893870441867,
0.21074069435558627893870441867, 1.68756826127567366277098844379, 1.94189992670889740080811254251, 2.95793528559440532826766385016, 3.82742139000405564426332305807, 4.64102308332650932047665131636, 4.69685100634529141342710886258, 5.48923003553805115382807850939, 5.58562915736206628470840563453, 6.95088234118917892805002856297, 6.99126396212536506782715607536, 7.44032182163567959265361623952, 8.061353232996184694510439069890, 8.542942690729029615989297886583, 8.836370075062313285927782482062, 9.635399519932249496501648038808, 9.971478098983298193625478616046, 10.77236707031257093073762291445, 10.79390716009387905853506156502, 11.61539604215763249739254180093
