| L(s) = 1 | + 6·3-s − 10·5-s + 2·7-s + 27·9-s − 22·11-s − 14·13-s − 60·15-s − 80·17-s − 60·19-s + 12·21-s + 202·23-s − 78·25-s + 108·27-s + 336·29-s − 128·31-s − 132·33-s − 20·35-s − 188·37-s − 84·39-s − 132·41-s + 480·43-s − 270·45-s + 390·47-s + 190·49-s − 480·51-s − 610·53-s + 220·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.107·7-s + 9-s − 0.603·11-s − 0.298·13-s − 1.03·15-s − 1.14·17-s − 0.724·19-s + 0.124·21-s + 1.83·23-s − 0.623·25-s + 0.769·27-s + 2.15·29-s − 0.741·31-s − 0.696·33-s − 0.0965·35-s − 0.835·37-s − 0.344·39-s − 0.502·41-s + 1.70·43-s − 0.894·45-s + 1.21·47-s + 0.553·49-s − 1.31·51-s − 1.58·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4460544 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.700062340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.700062340\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 p T + 178 T^{2} + 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 186 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 14 T - 310 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 80 T + 11038 T^{2} + 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 60 T + 4918 T^{2} + 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 202 T + 32110 T^{2} - 202 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 336 T + 73510 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 128 T + 38846 T^{2} + 128 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 188 T + 10814 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 132 T + 86326 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 480 T + 210406 T^{2} - 480 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 390 T + 244798 T^{2} - 390 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 610 T + 271954 T^{2} + 610 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 372 T + 413926 T^{2} - 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1050 T + 724834 T^{2} + 1050 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 408 T + 618310 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 942 T + 915838 T^{2} + 942 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 740 T + 517622 T^{2} - 740 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1646 T + 1592694 T^{2} - 1646 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 352 T + 912262 T^{2} + 352 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2036 T + 2421430 T^{2} - 2036 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 336 T + 1378270 T^{2} - 336 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.986519914073852983737115269820, −8.589485066840675898998354748900, −8.056846091011639402270225891112, −7.85488282972117486695863139093, −7.56593638650678868448290453685, −7.06979595060381390447593523413, −6.68618186105748232364384544395, −6.44480063297953212037032323474, −5.74131403365045712141317429271, −5.24269458836783584266721474748, −4.67730430899496427159819280776, −4.43858468326360981204370128778, −4.17350081562173339548578738042, −3.32296346156108713173120869426, −3.23553300184924927891854138400, −2.69354585719421119951053197634, −2.03206292638777303905789494311, −1.89269569597928656136053339211, −0.70252099637338597442032275270, −0.58420357474879079518219256285,
0.58420357474879079518219256285, 0.70252099637338597442032275270, 1.89269569597928656136053339211, 2.03206292638777303905789494311, 2.69354585719421119951053197634, 3.23553300184924927891854138400, 3.32296346156108713173120869426, 4.17350081562173339548578738042, 4.43858468326360981204370128778, 4.67730430899496427159819280776, 5.24269458836783584266721474748, 5.74131403365045712141317429271, 6.44480063297953212037032323474, 6.68618186105748232364384544395, 7.06979595060381390447593523413, 7.56593638650678868448290453685, 7.85488282972117486695863139093, 8.056846091011639402270225891112, 8.589485066840675898998354748900, 8.986519914073852983737115269820
