| L(s) = 1 | + 4·2-s + 12·4-s + 32·8-s + 80·16-s + 124·17-s + 168·19-s + 280·23-s + 32·31-s + 192·32-s + 496·34-s + 672·38-s + 1.12e3·46-s + 200·47-s − 190·49-s + 1.47e3·53-s − 716·61-s + 128·62-s + 448·64-s + 1.48e3·68-s + 2.01e3·76-s − 1.87e3·79-s + 2.60e3·83-s + 3.36e3·92-s + 800·94-s − 760·98-s + 5.90e3·106-s + 1.39e3·107-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 5/4·16-s + 1.76·17-s + 2.02·19-s + 2.53·23-s + 0.185·31-s + 1.06·32-s + 2.50·34-s + 2.86·38-s + 3.58·46-s + 0.620·47-s − 0.553·49-s + 3.82·53-s − 1.50·61-s + 0.262·62-s + 7/8·64-s + 2.65·68-s + 3.04·76-s − 2.66·79-s + 3.44·83-s + 3.80·92-s + 0.877·94-s − 0.783·98-s + 5.40·106-s + 1.25·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.49444170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.49444170\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 190 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2166 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 70 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 62 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 84 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 140 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 8602 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 41290 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 88242 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 32038 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 100 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 738 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6378 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 358 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 114698 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 159122 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 579634 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 936 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 1304 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 902034 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 1251970 T^{2} + p^{6} T^{4} \) |
| 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
−10.91325510212989642997366542315, −10.68718432887346367326811546760, −9.949048989581920647197480122687, −9.855758807957044587102758869553, −8.900879207728493049964041196471, −8.877068859602440315619710431198, −7.78759338580560266958756108244, −7.57343723924811754597840143077, −7.16902440058342457029628776373, −6.72075832164600314028619023790, −5.95557537195315445080649149502, −5.58037578033031578540529797356, −5.01034348210061453900338334788, −4.96465329895456523770127855772, −3.88596975781471889340555873069, −3.53573356912624628836371954998, −2.92567263138210704051459502448, −2.55143500753069135015772712354, −1.22000691428943146410391995160, −1.02322585538005756882312502793,
1.02322585538005756882312502793, 1.22000691428943146410391995160, 2.55143500753069135015772712354, 2.92567263138210704051459502448, 3.53573356912624628836371954998, 3.88596975781471889340555873069, 4.96465329895456523770127855772, 5.01034348210061453900338334788, 5.58037578033031578540529797356, 5.95557537195315445080649149502, 6.72075832164600314028619023790, 7.16902440058342457029628776373, 7.57343723924811754597840143077, 7.78759338580560266958756108244, 8.877068859602440315619710431198, 8.900879207728493049964041196471, 9.855758807957044587102758869553, 9.949048989581920647197480122687, 10.68718432887346367326811546760, 10.91325510212989642997366542315
