| L(s) = 1 | + 0.649·3-s − 2.92·5-s − 1.90·7-s − 2.57·9-s + 3.79·11-s + 1.26·13-s − 1.89·15-s − 3.64·17-s + 4.46·19-s − 1.23·21-s − 9.12·23-s + 3.54·25-s − 3.62·27-s + 2.77·29-s − 4.74·31-s + 2.46·33-s + 5.58·35-s + 1.24·37-s + 0.823·39-s + 8.26·41-s − 6.09·43-s + 7.53·45-s − 1.58·47-s − 3.35·49-s − 2.36·51-s + 9.70·53-s − 11.0·55-s + ⋯ |
| L(s) = 1 | + 0.374·3-s − 1.30·5-s − 0.721·7-s − 0.859·9-s + 1.14·11-s + 0.351·13-s − 0.490·15-s − 0.884·17-s + 1.02·19-s − 0.270·21-s − 1.90·23-s + 0.709·25-s − 0.697·27-s + 0.514·29-s − 0.853·31-s + 0.429·33-s + 0.943·35-s + 0.205·37-s + 0.131·39-s + 1.29·41-s − 0.929·43-s + 1.12·45-s − 0.230·47-s − 0.479·49-s − 0.331·51-s + 1.33·53-s − 1.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.063225250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063225250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.649T + 3T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 + 3.64T + 17T^{2} \) |
| 19 | \( 1 - 4.46T + 19T^{2} \) |
| 23 | \( 1 + 9.12T + 23T^{2} \) |
| 29 | \( 1 - 2.77T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 - 1.24T + 37T^{2} \) |
| 41 | \( 1 - 8.26T + 41T^{2} \) |
| 43 | \( 1 + 6.09T + 43T^{2} \) |
| 47 | \( 1 + 1.58T + 47T^{2} \) |
| 53 | \( 1 - 9.70T + 53T^{2} \) |
| 59 | \( 1 - 7.05T + 59T^{2} \) |
| 61 | \( 1 - 1.40T + 61T^{2} \) |
| 67 | \( 1 - 6.44T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 4.81T + 73T^{2} \) |
| 79 | \( 1 - 16.6T + 79T^{2} \) |
| 83 | \( 1 - 4.24T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 7.34T + 97T^{2} \) |
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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
−8.356480074997791375544965512301, −7.86906938524649884162739428478, −7.01162412356630211408392941966, −6.33067893574568232464271337278, −5.58112058339385921653750376746, −4.34425663885015667065651548757, −3.75883107302130759235069842991, −3.21991407224818294460325803963, −2.08314869346912362611484085817, −0.55829151816142679797004010049,
0.55829151816142679797004010049, 2.08314869346912362611484085817, 3.21991407224818294460325803963, 3.75883107302130759235069842991, 4.34425663885015667065651548757, 5.58112058339385921653750376746, 6.33067893574568232464271337278, 7.01162412356630211408392941966, 7.86906938524649884162739428478, 8.356480074997791375544965512301
