L(s) = 1 | − 3-s + 3.23·5-s − 0.763·7-s + 9-s − 4.47·11-s + 13-s − 3.23·15-s − 4.47·17-s + 3.23·19-s + 0.763·21-s − 6.47·23-s + 5.47·25-s − 27-s − 0.472·29-s − 4.76·31-s + 4.47·33-s − 2.47·35-s − 4.47·37-s − 39-s − 4.76·41-s − 2.47·43-s + 3.23·45-s − 8.47·47-s − 6.41·49-s + 4.47·51-s + 8.47·53-s − 14.4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.44·5-s − 0.288·7-s + 0.333·9-s − 1.34·11-s + 0.277·13-s − 0.835·15-s − 1.08·17-s + 0.742·19-s + 0.166·21-s − 1.34·23-s + 1.09·25-s − 0.192·27-s − 0.0876·29-s − 0.855·31-s + 0.778·33-s − 0.417·35-s − 0.735·37-s − 0.160·39-s − 0.744·41-s − 0.376·43-s + 0.482·45-s − 1.23·47-s − 0.916·49-s + 0.626·51-s + 1.16·53-s − 1.95·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 7 | \( 1 + 0.763T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 + 0.472T + 29T^{2} \) |
| 31 | \( 1 + 4.76T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 4.76T + 41T^{2} \) |
| 43 | \( 1 + 2.47T + 43T^{2} \) |
| 47 | \( 1 + 8.47T + 47T^{2} \) |
| 53 | \( 1 - 8.47T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 5.70T + 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 - 4.47T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 2.29T + 89T^{2} \) |
| 97 | \( 1 - 7.52T + 97T^{2} \) |
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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
−8.622743141324375129590504447764, −7.73514599627840100681766414843, −6.76977355717461548567420947736, −6.15588997456836058887207522205, −5.41540144659690839147801261852, −4.92264205118945584159750549089, −3.61349696638074206546179088610, −2.44512008565931364788640421093, −1.68975572598307039033093284927, 0,
1.68975572598307039033093284927, 2.44512008565931364788640421093, 3.61349696638074206546179088610, 4.92264205118945584159750549089, 5.41540144659690839147801261852, 6.15588997456836058887207522205, 6.76977355717461548567420947736, 7.73514599627840100681766414843, 8.622743141324375129590504447764