| L(s) = 1 | + 29·9-s − 78·11-s + 302·19-s − 384·29-s + 36·31-s + 458·41-s + 682·49-s − 672·59-s + 1.71e3·61-s + 1.56e3·71-s − 460·79-s + 112·81-s + 2.73e3·89-s − 2.26e3·99-s − 1.58e3·101-s − 892·109-s + 1.90e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.66e3·169-s + ⋯ |
| L(s) = 1 | + 1.07·9-s − 2.13·11-s + 3.64·19-s − 2.45·29-s + 0.208·31-s + 1.74·41-s + 1.98·49-s − 1.48·59-s + 3.60·61-s + 2.60·71-s − 0.655·79-s + 0.153·81-s + 3.26·89-s − 2.29·99-s − 1.56·101-s − 0.783·109-s + 1.42·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 1.21·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.695007583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.695007583\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 29 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 39 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 2662 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6105 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 151 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20970 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 192 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 18 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 82262 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 229 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 132118 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 162702 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 36330 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 336 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 858 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 557845 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 780 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 615625 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 230 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 528275 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1369 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1679422 T^{2} + p^{6} T^{4} \) |
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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
−11.12465080072562340770867535757, −10.58128241432074999638222300701, −9.990878595612598660703277764377, −9.918054300281964762245625564133, −9.209445740719670952106610329777, −9.103346589423949317988264741806, −7.910282377596835729813208521935, −7.85133730311744519298889962255, −7.29851259318978972731056174477, −7.28530267314271904419522932127, −6.30567275052962288945521705611, −5.50418318458000514619121336293, −5.21753486656324671773540152447, −5.12173810887201299868240399204, −3.88474608032378130037057038908, −3.70185621237209127973616967117, −2.73829305316966324934325726136, −2.33795150742758746190031377900, −1.26339501294110320429213289339, −0.60961234309670321078894777049,
0.60961234309670321078894777049, 1.26339501294110320429213289339, 2.33795150742758746190031377900, 2.73829305316966324934325726136, 3.70185621237209127973616967117, 3.88474608032378130037057038908, 5.12173810887201299868240399204, 5.21753486656324671773540152447, 5.50418318458000514619121336293, 6.30567275052962288945521705611, 7.28530267314271904419522932127, 7.29851259318978972731056174477, 7.85133730311744519298889962255, 7.910282377596835729813208521935, 9.103346589423949317988264741806, 9.209445740719670952106610329777, 9.918054300281964762245625564133, 9.990878595612598660703277764377, 10.58128241432074999638222300701, 11.12465080072562340770867535757
