| L(s) = 1 | + 5·5-s − 36·7-s + 64·11-s − 65·13-s − 59·17-s + 28·19-s + 160·23-s − 100·25-s + 57·29-s − 164·31-s − 180·35-s − 321·37-s + 246·41-s + 8·43-s + 84·47-s + 953·49-s − 478·53-s + 320·55-s − 32·59-s + 415·61-s − 325·65-s + 220·67-s + 884·71-s − 77·73-s − 2.30e3·77-s + 80·79-s + 1.26e3·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.94·7-s + 1.75·11-s − 1.38·13-s − 0.841·17-s + 0.338·19-s + 1.45·23-s − 4/5·25-s + 0.364·29-s − 0.950·31-s − 0.869·35-s − 1.42·37-s + 0.937·41-s + 0.0283·43-s + 0.260·47-s + 2.77·49-s − 1.23·53-s + 0.784·55-s − 0.0706·59-s + 0.871·61-s − 0.620·65-s + 0.401·67-s + 1.47·71-s − 0.123·73-s − 3.40·77-s + 0.113·79-s + 1.67·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.479134720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479134720\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - p T + p^{3} T^{2} \) |
| 7 | \( 1 + 36 T + p^{3} T^{2} \) |
| 11 | \( 1 - 64 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 59 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 160 T + p^{3} T^{2} \) |
| 29 | \( 1 - 57 T + p^{3} T^{2} \) |
| 31 | \( 1 + 164 T + p^{3} T^{2} \) |
| 37 | \( 1 + 321 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 8 T + p^{3} T^{2} \) |
| 47 | \( 1 - 84 T + p^{3} T^{2} \) |
| 53 | \( 1 + 478 T + p^{3} T^{2} \) |
| 59 | \( 1 + 32 T + p^{3} T^{2} \) |
| 61 | \( 1 - 415 T + p^{3} T^{2} \) |
| 67 | \( 1 - 220 T + p^{3} T^{2} \) |
| 71 | \( 1 - 884 T + p^{3} T^{2} \) |
| 73 | \( 1 + 77 T + p^{3} T^{2} \) |
| 79 | \( 1 - 80 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1268 T + p^{3} T^{2} \) |
| 89 | \( 1 + 123 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1346 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.390444754215999280304136826529, −8.894839520294054060600241709954, −7.30864176023285820194407224496, −6.75873887802125118430691598479, −6.19559833423199587343160817466, −5.12031169969128806570637992751, −3.93923633690027866528791559248, −3.15839967621550429549579989260, −2.07893947266411193565555522994, −0.59314301224210561529729410705,
0.59314301224210561529729410705, 2.07893947266411193565555522994, 3.15839967621550429549579989260, 3.93923633690027866528791559248, 5.12031169969128806570637992751, 6.19559833423199587343160817466, 6.75873887802125118430691598479, 7.30864176023285820194407224496, 8.894839520294054060600241709954, 9.390444754215999280304136826529
