L(s) = 1 | + 2.57·3-s − 5-s − 0.467·7-s + 3.64·9-s + 2.41·11-s − 4.23·13-s − 2.57·15-s + 2.57·17-s − 2.85·19-s − 1.20·21-s − 0.934·23-s + 25-s + 1.67·27-s + 2.53·29-s − 10.9·31-s + 6.23·33-s + 0.467·35-s − 8.90·37-s − 10.9·39-s − 3.09·41-s − 5.02·43-s − 3.64·45-s + 4.13·47-s − 6.78·49-s + 6.63·51-s − 4.20·53-s − 2.41·55-s + ⋯ |
L(s) = 1 | + 1.48·3-s − 0.447·5-s − 0.176·7-s + 1.21·9-s + 0.728·11-s − 1.17·13-s − 0.665·15-s + 0.624·17-s − 0.655·19-s − 0.263·21-s − 0.194·23-s + 0.200·25-s + 0.321·27-s + 0.470·29-s − 1.97·31-s + 1.08·33-s + 0.0790·35-s − 1.46·37-s − 1.75·39-s − 0.483·41-s − 0.765·43-s − 0.543·45-s + 0.602·47-s − 0.968·49-s + 0.929·51-s − 0.577·53-s − 0.325·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 2.57T + 3T^{2} \) |
| 7 | \( 1 + 0.467T + 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 + 2.85T + 19T^{2} \) |
| 23 | \( 1 + 0.934T + 23T^{2} \) |
| 29 | \( 1 - 2.53T + 29T^{2} \) |
| 31 | \( 1 + 10.9T + 31T^{2} \) |
| 37 | \( 1 + 8.90T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 + 5.02T + 43T^{2} \) |
| 47 | \( 1 - 4.13T + 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 59 | \( 1 - 1.69T + 59T^{2} \) |
| 61 | \( 1 + 6.17T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 - 9.23T + 71T^{2} \) |
| 73 | \( 1 - 1.95T + 73T^{2} \) |
| 79 | \( 1 - 6.90T + 79T^{2} \) |
| 83 | \( 1 - 5.25T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 7.86T + 97T^{2} \) |
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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
−7.50156548104903609688050935709, −7.12149018503638108393607880988, −6.28409262148069914519791715194, −5.22963736953712167156333350985, −4.48962288009534126457459246183, −3.58607596270089128388632283483, −3.29356317303386131737255005977, −2.26638881610648956002360103886, −1.60006364799957530216705937248, 0,
1.60006364799957530216705937248, 2.26638881610648956002360103886, 3.29356317303386131737255005977, 3.58607596270089128388632283483, 4.48962288009534126457459246183, 5.22963736953712167156333350985, 6.28409262148069914519791715194, 7.12149018503638108393607880988, 7.50156548104903609688050935709