| L(s) = 1 | + 4·4-s − 3·5-s + 7·7-s + 9·9-s − 13·13-s + 16·16-s + 25·19-s − 12·20-s − 45·23-s − 16·25-s + 28·28-s − 33·29-s − 55·31-s − 21·35-s + 36·36-s + 30·41-s − 5·43-s − 27·45-s + 81·47-s + 49·49-s − 52·52-s + 15·53-s − 90·59-s + 63·63-s + 64·64-s + 39·65-s + 29·73-s + ⋯ |
| L(s) = 1 | + 4-s − 3/5·5-s + 7-s + 9-s − 13-s + 16-s + 1.31·19-s − 3/5·20-s − 1.95·23-s − 0.639·25-s + 28-s − 1.13·29-s − 1.77·31-s − 3/5·35-s + 36-s + 0.731·41-s − 0.116·43-s − 3/5·45-s + 1.72·47-s + 49-s − 52-s + 0.283·53-s − 1.52·59-s + 63-s + 64-s + 3/5·65-s + 0.397·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.581817183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.581817183\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( ( 1 - p T )( 1 + p T ) \) |
| 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( 1 + 3 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 25 T + p^{2} T^{2} \) |
| 23 | \( 1 + 45 T + p^{2} T^{2} \) |
| 29 | \( 1 + 33 T + p^{2} T^{2} \) |
| 31 | \( 1 + 55 T + p^{2} T^{2} \) |
| 37 | \( ( 1 - p T )( 1 + p T ) \) |
| 41 | \( 1 - 30 T + p^{2} T^{2} \) |
| 43 | \( 1 + 5 T + p^{2} T^{2} \) |
| 47 | \( 1 - 81 T + p^{2} T^{2} \) |
| 53 | \( 1 - 15 T + p^{2} T^{2} \) |
| 59 | \( 1 + 90 T + p^{2} T^{2} \) |
| 61 | \( ( 1 - p T )( 1 + p T ) \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 29 T + p^{2} T^{2} \) |
| 79 | \( 1 - 67 T + p^{2} T^{2} \) |
| 83 | \( 1 + 159 T + p^{2} T^{2} \) |
| 89 | \( 1 - 165 T + p^{2} T^{2} \) |
| 97 | \( 1 + 131 T + p^{2} T^{2} \) |
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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
−14.03202004242769266745250894400, −12.41319049097375765724309959359, −11.77219191828574773977746987614, −10.77124065908922295565428080063, −9.614704084309157178729172855978, −7.67294963336278777803878616844, −7.42661843209808567462895054281, −5.58368608186619025512907937219, −3.98012363079663038800673054068, −1.91799578057070732308671927318,
1.91799578057070732308671927318, 3.98012363079663038800673054068, 5.58368608186619025512907937219, 7.42661843209808567462895054281, 7.67294963336278777803878616844, 9.614704084309157178729172855978, 10.77124065908922295565428080063, 11.77219191828574773977746987614, 12.41319049097375765724309959359, 14.03202004242769266745250894400
