| L(s) = 1 | + 1.61·3-s − 0.618·5-s + 7-s − 0.381·9-s + 11-s + 13-s − 1.00·15-s − 1.14·17-s + 2.61·19-s + 1.61·21-s − 7.70·23-s − 4.61·25-s − 5.47·27-s + 8.47·29-s − 7.23·31-s + 1.61·33-s − 0.618·35-s − 1.70·37-s + 1.61·39-s − 0.763·41-s − 0.381·43-s + 0.236·45-s − 9.23·47-s + 49-s − 1.85·51-s + 0.381·53-s − 0.618·55-s + ⋯ |
| L(s) = 1 | + 0.934·3-s − 0.276·5-s + 0.377·7-s − 0.127·9-s + 0.301·11-s + 0.277·13-s − 0.258·15-s − 0.277·17-s + 0.600·19-s + 0.353·21-s − 1.60·23-s − 0.923·25-s − 1.05·27-s + 1.57·29-s − 1.29·31-s + 0.281·33-s − 0.104·35-s − 0.280·37-s + 0.259·39-s − 0.119·41-s − 0.0582·43-s + 0.0351·45-s − 1.34·47-s + 0.142·49-s − 0.259·51-s + 0.0524·53-s − 0.0833·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 19 | \( 1 - 2.61T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 + 1.70T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 + 0.381T + 43T^{2} \) |
| 47 | \( 1 + 9.23T + 47T^{2} \) |
| 53 | \( 1 - 0.381T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 - 1.90T + 67T^{2} \) |
| 71 | \( 1 - 7.56T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 2.61T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.86979118746675367616008922768, −6.88437962990276544470056364993, −6.12620167658657621088943810951, −5.41682923818915039535062334747, −4.48771422911637910128307028216, −3.78040444640529188868115167518, −3.17687928372804894313245555221, −2.23829548766659474054653923799, −1.50521944352161166703004554658, 0,
1.50521944352161166703004554658, 2.23829548766659474054653923799, 3.17687928372804894313245555221, 3.78040444640529188868115167518, 4.48771422911637910128307028216, 5.41682923818915039535062334747, 6.12620167658657621088943810951, 6.88437962990276544470056364993, 7.86979118746675367616008922768
