| L(s) = 1 | + 3-s + 9-s − 11-s + 2·13-s − 6·17-s − 4·19-s − 4·23-s + 27-s − 6·29-s + 8·31-s − 33-s − 10·37-s + 2·39-s − 6·41-s + 4·43-s − 6·51-s + 6·53-s − 4·57-s − 2·61-s + 4·67-s − 4·69-s − 6·73-s + 4·79-s + 81-s − 6·87-s + 14·89-s + 8·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.301·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s − 0.834·23-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.174·33-s − 1.64·37-s + 0.320·39-s − 0.937·41-s + 0.609·43-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.256·61-s + 0.488·67-s − 0.481·69-s − 0.702·73-s + 0.450·79-s + 1/9·81-s − 0.643·87-s + 1.48·89-s + 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.03196028864598, −12.38769125389126, −11.97182640691935, −11.50230337948858, −10.92439077396621, −10.55933326739773, −10.16570905912899, −9.661228944639955, −8.957831294862970, −8.770110830395587, −8.356238700819206, −7.841870836762407, −7.323357536855067, −6.750962904002935, −6.403657946979204, −5.905249536193179, −5.258253688265032, −4.687992240637243, −4.229622769284870, −3.746744069652096, −3.252244118236370, −2.528555031110769, −2.033390705215763, −1.701622051966356, −0.7039983922496206, 0,
0.7039983922496206, 1.701622051966356, 2.033390705215763, 2.528555031110769, 3.252244118236370, 3.746744069652096, 4.229622769284870, 4.687992240637243, 5.258253688265032, 5.905249536193179, 6.403657946979204, 6.750962904002935, 7.323357536855067, 7.841870836762407, 8.356238700819206, 8.770110830395587, 8.957831294862970, 9.661228944639955, 10.16570905912899, 10.55933326739773, 10.92439077396621, 11.50230337948858, 11.97182640691935, 12.38769125389126, 13.03196028864598
