| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 2·7-s − 8-s + 10-s − 10·11-s − 5·13-s − 2·14-s + 15-s − 16-s + 5·17-s − 2·19-s − 2·21-s − 10·22-s − 24-s − 5·26-s − 27-s − 18·29-s + 30-s + 8·31-s − 10·33-s + 5·34-s − 2·35-s + 37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 0.316·10-s − 3.01·11-s − 1.38·13-s − 0.534·14-s + 0.258·15-s − 1/4·16-s + 1.21·17-s − 0.458·19-s − 0.436·21-s − 2.13·22-s − 0.204·24-s − 0.980·26-s − 0.192·27-s − 3.34·29-s + 0.182·30-s + 1.43·31-s − 1.74·33-s + 0.857·34-s − 0.338·35-s + 0.164·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.132181111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.132181111\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 37 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 9 T + 14 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 12 T + 73 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 16 T + 177 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 2 T - 79 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−9.895229605154386705753371611054, −9.830617483174062762974913241447, −9.258885623341698062529608626560, −8.952677386355150180209916458371, −8.279524988625482730407823827717, −7.79388202743721958840015857397, −7.67797784157869321364174352747, −7.27202117645734074763766748881, −6.79331068334586798801399775171, −5.92412104781530460098390377473, −5.67999674593804376707799382122, −5.25889641366113415656529714713, −5.22187997594200934365590020570, −4.17533733727464206227780713276, −4.07700134980005773011811315758, −3.01787900420208888078971942426, −2.91216122600353537452807139657, −2.46533774132156703503218947021, −1.90660766179344916812031405289, −0.36968190955454456474643329672,
0.36968190955454456474643329672, 1.90660766179344916812031405289, 2.46533774132156703503218947021, 2.91216122600353537452807139657, 3.01787900420208888078971942426, 4.07700134980005773011811315758, 4.17533733727464206227780713276, 5.22187997594200934365590020570, 5.25889641366113415656529714713, 5.67999674593804376707799382122, 5.92412104781530460098390377473, 6.79331068334586798801399775171, 7.27202117645734074763766748881, 7.67797784157869321364174352747, 7.79388202743721958840015857397, 8.279524988625482730407823827717, 8.952677386355150180209916458371, 9.258885623341698062529608626560, 9.830617483174062762974913241447, 9.895229605154386705753371611054
