L(s) = 1 | + (1 − 1.73i)2-s + (−0.5 − 0.866i)3-s + (−1.99 − 3.46i)4-s + (3.5 − 6.06i)5-s − 1.99·6-s − 7.99·8-s + (13 − 22.5i)9-s + (−7 − 12.1i)10-s + (−17.5 − 30.3i)11-s + (−1.99 + 3.46i)12-s − 66·13-s − 7·15-s + (−8 + 13.8i)16-s + (29.5 + 51.0i)17-s + (−26 − 45.0i)18-s + (68.5 − 118. i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.0962 − 0.166i)3-s + (−0.249 − 0.433i)4-s + (0.313 − 0.542i)5-s − 0.136·6-s − 0.353·8-s + (0.481 − 0.833i)9-s + (−0.221 − 0.383i)10-s + (−0.479 − 0.830i)11-s + (−0.0481 + 0.0833i)12-s − 1.40·13-s − 0.120·15-s + (−0.125 + 0.216i)16-s + (0.420 + 0.728i)17-s + (−0.340 − 0.589i)18-s + (0.827 − 1.43i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.729474 - 1.47152i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.729474 - 1.47152i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 + (-3.5 + 6.06i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (17.5 + 30.3i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 66T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-29.5 - 51.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-68.5 + 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-3.5 + 6.06i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 106T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-37.5 - 64.9i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (5.5 - 9.52i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 498T + 6.89e4T^{2} \) |
| 43 | \( 1 - 260T + 7.95e4T^{2} \) |
| 47 | \( 1 + (85.5 - 148. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-208.5 - 361. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (8.5 + 14.7i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-25.5 + 44.1i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (219.5 + 380. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 784T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-147.5 - 255. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-247.5 + 428. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 932T + 5.71e5T^{2} \) |
| 89 | \( 1 + (436.5 - 756. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 290T + 9.12e5T^{2} \) |
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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
−12.85164898889506516891526159190, −12.23304108328036510648978664059, −11.04352786611972326818903308025, −9.825213500095759776591501489114, −8.958782792089562890978329087004, −7.33146557393786375184961152936, −5.81549740700953971713696179583, −4.62028414642847223046156168397, −2.89144620221992818889475911496, −0.896018045867936731791782438673,
2.54156757359145915234473106195, 4.51651740539359137357503085798, 5.57830663988827454210390884204, 7.16358221752932503464067860583, 7.80806810345649549707131232111, 9.701020819243641939159994261177, 10.32716730388733783864823702041, 11.92991772655909124463845465725, 12.83754641785020293197941347850, 14.08871258689931858751094523211