L(s) = 1 | + 2-s − 2.35·3-s + 4-s + 0.827·5-s − 2.35·6-s + 0.166·7-s + 8-s + 2.53·9-s + 0.827·10-s − 5.71·11-s − 2.35·12-s − 3.76·13-s + 0.166·14-s − 1.94·15-s + 16-s + 1.08·17-s + 2.53·18-s − 7.95·19-s + 0.827·20-s − 0.392·21-s − 5.71·22-s + 23-s − 2.35·24-s − 4.31·25-s − 3.76·26-s + 1.09·27-s + 0.166·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.35·3-s + 0.5·4-s + 0.369·5-s − 0.960·6-s + 0.0630·7-s + 0.353·8-s + 0.844·9-s + 0.261·10-s − 1.72·11-s − 0.679·12-s − 1.04·13-s + 0.0445·14-s − 0.502·15-s + 0.250·16-s + 0.262·17-s + 0.596·18-s − 1.82·19-s + 0.184·20-s − 0.0855·21-s − 1.21·22-s + 0.208·23-s − 0.480·24-s − 0.863·25-s − 0.737·26-s + 0.211·27-s + 0.0315·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.078940464\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.078940464\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 - 0.827T + 5T^{2} \) |
| 7 | \( 1 - 0.166T + 7T^{2} \) |
| 11 | \( 1 + 5.71T + 11T^{2} \) |
| 13 | \( 1 + 3.76T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 + 7.95T + 19T^{2} \) |
| 29 | \( 1 + 4.77T + 29T^{2} \) |
| 31 | \( 1 - 2.25T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 4.45T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 10.2T + 53T^{2} \) |
| 59 | \( 1 + 3.08T + 59T^{2} \) |
| 61 | \( 1 - 5.61T + 61T^{2} \) |
| 67 | \( 1 - 0.392T + 67T^{2} \) |
| 71 | \( 1 + 3.94T + 71T^{2} \) |
| 73 | \( 1 - 8.69T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 5.44T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.78145482504005733741184312209, −7.26751483146678808445357336039, −6.23686625902091226784750181639, −5.98353470279077021292228145652, −5.07190375761233289923843903677, −4.86514680117529180219197602045, −3.89575223604621707193379018770, −2.60615096194360918144474892183, −2.10066416188724797023956006315, −0.49171288198688399515582741711,
0.49171288198688399515582741711, 2.10066416188724797023956006315, 2.60615096194360918144474892183, 3.89575223604621707193379018770, 4.86514680117529180219197602045, 5.07190375761233289923843903677, 5.98353470279077021292228145652, 6.23686625902091226784750181639, 7.26751483146678808445357336039, 7.78145482504005733741184312209