| L(s) = 1 | + 2-s + 5·3-s − 11·4-s − 3·5-s + 5·6-s − 9·7-s − 15·8-s + 3·9-s − 3·10-s + 80·11-s − 55·12-s − 26·13-s − 9·14-s − 15·15-s + 61·16-s + 19·17-s + 3·18-s − 84·19-s + 33·20-s − 45·21-s + 80·22-s + 196·23-s − 75·24-s − 239·25-s − 26·26-s + 40·27-s + 99·28-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.962·3-s − 1.37·4-s − 0.268·5-s + 0.340·6-s − 0.485·7-s − 0.662·8-s + 1/9·9-s − 0.0948·10-s + 2.19·11-s − 1.32·12-s − 0.554·13-s − 0.171·14-s − 0.258·15-s + 0.953·16-s + 0.271·17-s + 0.0392·18-s − 1.01·19-s + 0.368·20-s − 0.467·21-s + 0.775·22-s + 1.77·23-s − 0.637·24-s − 1.91·25-s − 0.196·26-s + 0.285·27-s + 0.668·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.009426291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.009426291\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 5 T + 22 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 248 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 9 T + 192 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 80 T + 3650 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 19 T + 8688 T^{2} - 19 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 84 T + 11130 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 196 T + 33326 T^{2} - 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 44 T + 10094 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 86 T + 56518 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 209 T + 112120 T^{2} - 209 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 230 T + 149010 T^{2} + 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 287 T + 92698 T^{2} - 287 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 435 T + 192728 T^{2} - 435 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 118 T + 297410 T^{2} + 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 368 T + 379266 T^{2} + 368 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1058 T + 580378 T^{2} + 1058 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 68 T + 373930 T^{2} - 68 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 131 T + 493328 T^{2} + 131 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 456 T + 542718 T^{2} - 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1008 T + 1233294 T^{2} + 1008 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1958 T + 1961238 T^{2} - 1958 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 720 T + 899726 T^{2} + 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 928 T + 943870 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−19.60374200828868102468536634216, −19.30705881950751410189199659154, −18.77683236558196979258764124750, −17.76737342376482072933577699954, −17.04482603597956831823514602140, −16.75650718689160499623937765796, −15.24718868092760447281081961144, −14.90239701884330161535029855708, −14.05008445734507485159626055694, −13.87746652659443314443937241466, −12.85250954284156864464523605367, −12.28434038437570960343096293189, −11.26038422060788727187354048575, −9.827288916794965173808830426038, −9.046158984158336058320394800623, −8.882764193305829362779320448505, −7.53141004237313528585943938051, −6.17812286510206894290657191844, −4.48460182468764722408570928051, −3.56763813396559653157210978973,
3.56763813396559653157210978973, 4.48460182468764722408570928051, 6.17812286510206894290657191844, 7.53141004237313528585943938051, 8.882764193305829362779320448505, 9.046158984158336058320394800623, 9.827288916794965173808830426038, 11.26038422060788727187354048575, 12.28434038437570960343096293189, 12.85250954284156864464523605367, 13.87746652659443314443937241466, 14.05008445734507485159626055694, 14.90239701884330161535029855708, 15.24718868092760447281081961144, 16.75650718689160499623937765796, 17.04482603597956831823514602140, 17.76737342376482072933577699954, 18.77683236558196979258764124750, 19.30705881950751410189199659154, 19.60374200828868102468536634216
