| L(s) = 1 | + 2.69·2-s + 5.27·4-s − 3.56·7-s + 8.84·8-s + 0.0695·11-s + 4.85·13-s − 9.62·14-s + 13.2·16-s − 3.03·17-s + 5.05·19-s + 0.187·22-s + 7.43·23-s + 13.1·26-s − 18.8·28-s − 1.95·29-s − 5.63·31-s + 18.1·32-s − 8.18·34-s + 6.21·37-s + 13.6·38-s + 5.63·41-s + 0.244·43-s + 0.367·44-s + 20.0·46-s + 3.23·47-s + 5.71·49-s + 25.6·52-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.63·4-s − 1.34·7-s + 3.12·8-s + 0.0209·11-s + 1.34·13-s − 2.57·14-s + 3.32·16-s − 0.735·17-s + 1.15·19-s + 0.0400·22-s + 1.54·23-s + 2.56·26-s − 3.55·28-s − 0.362·29-s − 1.01·31-s + 3.21·32-s − 1.40·34-s + 1.02·37-s + 2.21·38-s + 0.880·41-s + 0.0372·43-s + 0.0553·44-s + 2.95·46-s + 0.471·47-s + 0.817·49-s + 3.55·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.033216027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.033216027\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.69T + 2T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 - 0.0695T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 + 3.03T + 17T^{2} \) |
| 19 | \( 1 - 5.05T + 19T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 + 1.95T + 29T^{2} \) |
| 31 | \( 1 + 5.63T + 31T^{2} \) |
| 37 | \( 1 - 6.21T + 37T^{2} \) |
| 41 | \( 1 - 5.63T + 41T^{2} \) |
| 43 | \( 1 - 0.244T + 43T^{2} \) |
| 47 | \( 1 - 3.23T + 47T^{2} \) |
| 53 | \( 1 + 8.37T + 53T^{2} \) |
| 59 | \( 1 - 1.60T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + 2.94T + 67T^{2} \) |
| 71 | \( 1 + 7.25T + 71T^{2} \) |
| 73 | \( 1 - 3.69T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 - 8.80T + 83T^{2} \) |
| 89 | \( 1 - 3.55T + 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−7.69006625518857706959352140594, −7.08772893117425612995840661687, −6.40533163036203031814756148201, −5.97205678377160814987097869544, −5.25626728395564686849559053728, −4.42253486124325983163595424682, −3.55858900723518093646898890800, −3.23932246192584241601747611207, −2.36929910306360820825497038116, −1.12125266332879329012240768042,
1.12125266332879329012240768042, 2.36929910306360820825497038116, 3.23932246192584241601747611207, 3.55858900723518093646898890800, 4.42253486124325983163595424682, 5.25626728395564686849559053728, 5.97205678377160814987097869544, 6.40533163036203031814756148201, 7.08772893117425612995840661687, 7.69006625518857706959352140594
