L(s) = 1 | + 1.36·3-s + 0.636·7-s − 1.14·9-s − 3.50·11-s + 0.141·13-s − 2.14·17-s − 19-s + 0.867·21-s + 4.91·23-s − 5.64·27-s + 7.15·29-s + 7.78·31-s − 4.77·33-s − 3.27·37-s + 0.192·39-s − 4.23·41-s + 2.49·43-s − 10.2·47-s − 6.59·49-s − 2.91·51-s − 8.14·53-s − 1.36·57-s + 5.64·59-s − 6.49·61-s − 0.726·63-s + 8.37·67-s + 6.70·69-s + ⋯ |
L(s) = 1 | + 0.787·3-s + 0.240·7-s − 0.380·9-s − 1.05·11-s + 0.0391·13-s − 0.519·17-s − 0.229·19-s + 0.189·21-s + 1.02·23-s − 1.08·27-s + 1.32·29-s + 1.39·31-s − 0.831·33-s − 0.538·37-s + 0.0308·39-s − 0.660·41-s + 0.380·43-s − 1.49·47-s − 0.942·49-s − 0.408·51-s − 1.11·53-s − 0.180·57-s + 0.735·59-s − 0.831·61-s − 0.0915·63-s + 1.02·67-s + 0.807·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 1.36T + 3T^{2} \) |
| 7 | \( 1 - 0.636T + 7T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 - 0.141T + 13T^{2} \) |
| 17 | \( 1 + 2.14T + 17T^{2} \) |
| 23 | \( 1 - 4.91T + 23T^{2} \) |
| 29 | \( 1 - 7.15T + 29T^{2} \) |
| 31 | \( 1 - 7.78T + 31T^{2} \) |
| 37 | \( 1 + 3.27T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 - 2.49T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 8.14T + 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 - 8.37T + 67T^{2} \) |
| 71 | \( 1 - 8.95T + 71T^{2} \) |
| 73 | \( 1 - 3.69T + 73T^{2} \) |
| 79 | \( 1 - 4.17T + 79T^{2} \) |
| 83 | \( 1 + 9.00T + 83T^{2} \) |
| 89 | \( 1 + 6.77T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.79706788353094037584022539273, −6.80491765262238080265663205462, −6.30927681820060854114535486773, −5.17147753920808443339533081525, −4.85934422797038473989352332428, −3.80364338301390651296048300171, −2.87576342691802929154286373239, −2.55439581477158978550842357845, −1.40150388105516658163866047025, 0,
1.40150388105516658163866047025, 2.55439581477158978550842357845, 2.87576342691802929154286373239, 3.80364338301390651296048300171, 4.85934422797038473989352332428, 5.17147753920808443339533081525, 6.30927681820060854114535486773, 6.80491765262238080265663205462, 7.79706788353094037584022539273