L(s) = 1 | − 3-s + 2.79·5-s − 4.56·7-s + 9-s − 2.91·11-s − 0.850·13-s − 2.79·15-s − 7.79·17-s − 2.29·19-s + 4.56·21-s − 2.74·23-s + 2.82·25-s − 27-s − 6.57·29-s + 7.06·31-s + 2.91·33-s − 12.7·35-s + 8.35·37-s + 0.850·39-s + 4.96·41-s − 11.4·43-s + 2.79·45-s − 12.1·47-s + 13.8·49-s + 7.79·51-s − 6.03·53-s − 8.14·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.25·5-s − 1.72·7-s + 0.333·9-s − 0.877·11-s − 0.235·13-s − 0.722·15-s − 1.89·17-s − 0.527·19-s + 0.996·21-s − 0.572·23-s + 0.564·25-s − 0.192·27-s − 1.22·29-s + 1.26·31-s + 0.506·33-s − 2.15·35-s + 1.37·37-s + 0.136·39-s + 0.776·41-s − 1.74·43-s + 0.416·45-s − 1.77·47-s + 1.97·49-s + 1.09·51-s − 0.829·53-s − 1.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7229340163\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7229340163\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 - 2.79T + 5T^{2} \) |
| 7 | \( 1 + 4.56T + 7T^{2} \) |
| 11 | \( 1 + 2.91T + 11T^{2} \) |
| 13 | \( 1 + 0.850T + 13T^{2} \) |
| 17 | \( 1 + 7.79T + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 - 7.06T + 31T^{2} \) |
| 37 | \( 1 - 8.35T + 37T^{2} \) |
| 41 | \( 1 - 4.96T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 6.03T + 53T^{2} \) |
| 59 | \( 1 - 3.64T + 59T^{2} \) |
| 61 | \( 1 - 3.09T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 - 0.961T + 71T^{2} \) |
| 73 | \( 1 + 6.57T + 73T^{2} \) |
| 79 | \( 1 + 2.76T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 8.20T + 89T^{2} \) |
| 97 | \( 1 + 8.44T + 97T^{2} \) |
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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
−7.74744568311671421520695976164, −6.73458604335190248523800127348, −6.39719176929711087656066073070, −5.99944811451290624671689223552, −5.13758041078028170865823729202, −4.43245071038647538600224078140, −3.42991216057959171532238050636, −2.49198347513729567209358991390, −1.99451214026275215599294297694, −0.40123076677850108841055923813,
0.40123076677850108841055923813, 1.99451214026275215599294297694, 2.49198347513729567209358991390, 3.42991216057959171532238050636, 4.43245071038647538600224078140, 5.13758041078028170865823729202, 5.99944811451290624671689223552, 6.39719176929711087656066073070, 6.73458604335190248523800127348, 7.74744568311671421520695976164