| L(s) = 1 | − 2-s + 5-s + 7-s + 8-s − 10-s − 5·13-s − 14-s − 16-s + 12·17-s + 10·19-s + 3·23-s + 5·26-s − 8·31-s − 12·34-s + 35-s + 4·37-s − 10·38-s + 40-s + 3·41-s + 4·43-s − 3·46-s + 9·47-s + 7·49-s + 18·53-s + 56-s + 15·59-s + 4·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 1.38·13-s − 0.267·14-s − 1/4·16-s + 2.91·17-s + 2.29·19-s + 0.625·23-s + 0.980·26-s − 1.43·31-s − 2.05·34-s + 0.169·35-s + 0.657·37-s − 1.62·38-s + 0.158·40-s + 0.468·41-s + 0.609·43-s − 0.442·46-s + 1.31·47-s + 49-s + 2.47·53-s + 0.133·56-s + 1.95·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.833778794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.833778794\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−10.05806407696780296353680003420, −9.992253968358053581709642170429, −9.683089258214469102317002500048, −9.425406196563417174862777579762, −8.752565747990412956295563000507, −8.441231889834241410444865536492, −7.74633544233020520932982400496, −7.46880445712703811001242648754, −7.25590533988163258854165038991, −6.93893245179430259196424098741, −5.65288655609401450071904867945, −5.58623570113554852034371734143, −5.44518027554568660016134346260, −4.80796668008109722841891914871, −3.90107390501358657232797877074, −3.60936512599670516208090366324, −2.71722313015132763593091342164, −2.41472783484015446598791664057, −1.18947032585868121879335175876, −0.966553333684992349034043925735,
0.966553333684992349034043925735, 1.18947032585868121879335175876, 2.41472783484015446598791664057, 2.71722313015132763593091342164, 3.60936512599670516208090366324, 3.90107390501358657232797877074, 4.80796668008109722841891914871, 5.44518027554568660016134346260, 5.58623570113554852034371734143, 5.65288655609401450071904867945, 6.93893245179430259196424098741, 7.25590533988163258854165038991, 7.46880445712703811001242648754, 7.74633544233020520932982400496, 8.441231889834241410444865536492, 8.752565747990412956295563000507, 9.425406196563417174862777579762, 9.683089258214469102317002500048, 9.992253968358053581709642170429, 10.05806407696780296353680003420
