| L(s) = 1 | − 9·9-s + 72·11-s + 152·19-s − 108·29-s − 224·31-s + 756·41-s − 49·49-s − 792·59-s + 508·61-s + 1.68e3·71-s − 160·79-s + 81·81-s + 3.27e3·89-s − 648·99-s + 12·101-s + 3.57e3·109-s + 1.22e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 550·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 1.97·11-s + 1.83·19-s − 0.691·29-s − 1.29·31-s + 2.87·41-s − 1/7·49-s − 1.74·59-s + 1.06·61-s + 2.80·71-s − 0.227·79-s + 1/9·81-s + 3.90·89-s − 0.657·99-s + 0.0118·101-s + 3.13·109-s + 0.921·121-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 + 0.250·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.343946372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.343946372\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 550 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3170 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 23758 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 54 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 69622 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 378 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 p^{2} T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 170782 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 136150 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 396 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 254 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 422618 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 840 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 14066 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 80 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1131910 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1638 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 805246 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.100071637183052757894521826441, −8.725498138056501790108984920987, −8.095656822943520235798063752343, −7.60310265203676842661261342797, −7.53716026742163664876792180521, −7.05662556515025833402253253798, −6.49792785027720709524467309100, −6.25804013869548601303013623160, −5.78509162359177582403696286598, −5.46291260066862729495207765356, −4.90806321176232284564947806113, −4.53731981394652472046129049001, −3.82451128061226849235444397028, −3.67606237143979900683740851783, −3.29673543950176231738164609883, −2.59221308633659153199323710655, −2.02645294770341737065295750505, −1.52715254308672523505901049761, −0.840370260227885977052932726377, −0.62072508189128818108380272740,
0.62072508189128818108380272740, 0.840370260227885977052932726377, 1.52715254308672523505901049761, 2.02645294770341737065295750505, 2.59221308633659153199323710655, 3.29673543950176231738164609883, 3.67606237143979900683740851783, 3.82451128061226849235444397028, 4.53731981394652472046129049001, 4.90806321176232284564947806113, 5.46291260066862729495207765356, 5.78509162359177582403696286598, 6.25804013869548601303013623160, 6.49792785027720709524467309100, 7.05662556515025833402253253798, 7.53716026742163664876792180521, 7.60310265203676842661261342797, 8.095656822943520235798063752343, 8.725498138056501790108984920987, 9.100071637183052757894521826441
