L(s) = 1 | − 3-s − 4.15·5-s − 1.11·7-s + 9-s − 5.32·11-s + 0.315·13-s + 4.15·15-s − 1.36·17-s + 5.63·19-s + 1.11·21-s + 23-s + 12.2·25-s − 27-s − 29-s − 7.05·31-s + 5.32·33-s + 4.63·35-s − 5.05·37-s − 0.315·39-s + 2.34·41-s + 4.48·43-s − 4.15·45-s + 3.25·47-s − 5.75·49-s + 1.36·51-s + 7.64·53-s + 22.1·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.85·5-s − 0.421·7-s + 0.333·9-s − 1.60·11-s + 0.0875·13-s + 1.07·15-s − 0.331·17-s + 1.29·19-s + 0.243·21-s + 0.208·23-s + 2.45·25-s − 0.192·27-s − 0.185·29-s − 1.26·31-s + 0.926·33-s + 0.782·35-s − 0.831·37-s − 0.0505·39-s + 0.366·41-s + 0.683·43-s − 0.619·45-s + 0.474·47-s − 0.822·49-s + 0.191·51-s + 1.05·53-s + 2.98·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
good | 5 | \( 1 + 4.15T + 5T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 + 5.32T + 11T^{2} \) |
| 13 | \( 1 - 0.315T + 13T^{2} \) |
| 17 | \( 1 + 1.36T + 17T^{2} \) |
| 19 | \( 1 - 5.63T + 19T^{2} \) |
| 31 | \( 1 + 7.05T + 31T^{2} \) |
| 37 | \( 1 + 5.05T + 37T^{2} \) |
| 41 | \( 1 - 2.34T + 41T^{2} \) |
| 43 | \( 1 - 4.48T + 43T^{2} \) |
| 47 | \( 1 - 3.25T + 47T^{2} \) |
| 53 | \( 1 - 7.64T + 53T^{2} \) |
| 59 | \( 1 + 5.11T + 59T^{2} \) |
| 61 | \( 1 - 1.44T + 61T^{2} \) |
| 67 | \( 1 - 1.32T + 67T^{2} \) |
| 71 | \( 1 - 5.18T + 71T^{2} \) |
| 73 | \( 1 - 2.17T + 73T^{2} \) |
| 79 | \( 1 - 16.7T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 7.27T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−7.50822380097024663176854950533, −7.06775866393348485055934950011, −6.09704104679721474706512571789, −5.15162039295950633219762665301, −4.85925537850983143515617952089, −3.74665097988547237566769199685, −3.37804367751554691220155940984, −2.37223667776865014923104019748, −0.807268740114146943614618514108, 0,
0.807268740114146943614618514108, 2.37223667776865014923104019748, 3.37804367751554691220155940984, 3.74665097988547237566769199685, 4.85925537850983143515617952089, 5.15162039295950633219762665301, 6.09704104679721474706512571789, 7.06775866393348485055934950011, 7.50822380097024663176854950533