| L(s) = 1 | − 3-s + 0.618·7-s + 9-s − 3·11-s + 3·13-s − 1.85·17-s + 0.236·19-s − 0.618·21-s − 23-s − 27-s − 8.70·29-s − 1.38·31-s + 3·33-s + 5·37-s − 3·39-s + 5.09·41-s + 1.14·43-s − 9.47·47-s − 6.61·49-s + 1.85·51-s + 12.5·53-s − 0.236·57-s − 9.09·59-s + 2.32·61-s + 0.618·63-s + 8·67-s + 69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.233·7-s + 0.333·9-s − 0.904·11-s + 0.832·13-s − 0.449·17-s + 0.0541·19-s − 0.134·21-s − 0.208·23-s − 0.192·27-s − 1.61·29-s − 0.248·31-s + 0.522·33-s + 0.821·37-s − 0.480·39-s + 0.794·41-s + 0.174·43-s − 1.38·47-s − 0.945·49-s + 0.259·51-s + 1.72·53-s − 0.0312·57-s − 1.18·59-s + 0.297·61-s + 0.0778·63-s + 0.977·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + 1.85T + 17T^{2} \) |
| 19 | \( 1 - 0.236T + 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 - 5.09T + 41T^{2} \) |
| 43 | \( 1 - 1.14T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 9.09T + 59T^{2} \) |
| 61 | \( 1 - 2.32T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 3.14T + 71T^{2} \) |
| 73 | \( 1 + 0.381T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 + 5.38T + 83T^{2} \) |
| 89 | \( 1 + 3.47T + 89T^{2} \) |
| 97 | \( 1 - 0.618T + 97T^{2} \) |
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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
−8.254777884451253461837492894245, −7.63418460299726467343764435523, −6.81939465447280868039880363710, −5.93688146102876986082494030169, −5.40542832447722486977301417196, −4.49074337003268988208339441893, −3.67423346327403702267435335864, −2.52312433960917950022670243840, −1.42855235278323048841428005593, 0,
1.42855235278323048841428005593, 2.52312433960917950022670243840, 3.67423346327403702267435335864, 4.49074337003268988208339441893, 5.40542832447722486977301417196, 5.93688146102876986082494030169, 6.81939465447280868039880363710, 7.63418460299726467343764435523, 8.254777884451253461837492894245
