| L(s) = 1 | − 4-s − 9-s + 8·11-s + 16-s − 2·19-s + 6·29-s + 8·31-s + 36-s + 18·41-s − 8·44-s + 14·49-s − 20·59-s + 8·61-s − 64-s + 14·71-s + 2·76-s − 22·79-s + 81-s − 20·89-s − 8·99-s − 4·101-s + 2·109-s − 6·116-s + 26·121-s − 8·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 2.41·11-s + 1/4·16-s − 0.458·19-s + 1.11·29-s + 1.43·31-s + 1/6·36-s + 2.81·41-s − 1.20·44-s + 2·49-s − 2.60·59-s + 1.02·61-s − 1/8·64-s + 1.66·71-s + 0.229·76-s − 2.47·79-s + 1/9·81-s − 2.11·89-s − 0.804·99-s − 0.398·101-s + 0.191·109-s − 0.557·116-s + 2.36·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.759850295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.759850295\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−9.383959773662491739088867457866, −8.963679514230974321360070723235, −8.665187066110246081918010748592, −8.348238410856086556441566436439, −7.931159534036808407455438329429, −7.35597352156826175290457893333, −7.00775961945748723362853757909, −6.54601329253037843933164509752, −6.24690587688945359925144397871, −5.83981990162848789514207884289, −5.53235116207398038132721084147, −4.62373473455971165655363489474, −4.51991499107931442992903470554, −3.91697028591255359560411726887, −3.90818494239866759062044863943, −2.82140698208012747522960512116, −2.80127935973317851593710413744, −1.83732497093286792182380406742, −1.15483621915876555702355893796, −0.72773790014153021260358001658,
0.72773790014153021260358001658, 1.15483621915876555702355893796, 1.83732497093286792182380406742, 2.80127935973317851593710413744, 2.82140698208012747522960512116, 3.90818494239866759062044863943, 3.91697028591255359560411726887, 4.51991499107931442992903470554, 4.62373473455971165655363489474, 5.53235116207398038132721084147, 5.83981990162848789514207884289, 6.24690587688945359925144397871, 6.54601329253037843933164509752, 7.00775961945748723362853757909, 7.35597352156826175290457893333, 7.931159534036808407455438329429, 8.348238410856086556441566436439, 8.665187066110246081918010748592, 8.963679514230974321360070723235, 9.383959773662491739088867457866
