L(s) = 1 | + (1.76 + 1.37i)5-s + (−4.27 − 2.46i)7-s + (1.20 − 2.08i)11-s + (−2.51 + 1.45i)13-s + 6.86i·17-s + 4.17·19-s + (−2.90 + 1.67i)23-s + (1.21 + 4.84i)25-s + (2.59 − 4.5i)29-s + (3.08 + 5.35i)31-s + (−4.14 − 10.2i)35-s + 7.84i·37-s + (2.93 + 5.08i)41-s + (4.27 + 2.46i)43-s + (10.3 + 5.95i)47-s + ⋯ |
L(s) = 1 | + (0.788 + 0.614i)5-s + (−1.61 − 0.932i)7-s + (0.363 − 0.629i)11-s + (−0.698 + 0.403i)13-s + 1.66i·17-s + 0.958·19-s + (−0.606 + 0.350i)23-s + (0.243 + 0.969i)25-s + (0.482 − 0.835i)29-s + (0.554 + 0.961i)31-s + (−0.700 − 1.72i)35-s + 1.28i·37-s + (0.458 + 0.794i)41-s + (0.651 + 0.376i)43-s + (1.50 + 0.868i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.347852886\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.347852886\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.76 - 1.37i)T \) |
good | 7 | \( 1 + (4.27 + 2.46i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.20 + 2.08i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.51 - 1.45i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 6.86iT - 17T^{2} \) |
| 19 | \( 1 - 4.17T + 19T^{2} \) |
| 23 | \( 1 + (2.90 - 1.67i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.59 + 4.5i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.08 - 5.35i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.84iT - 37T^{2} \) |
| 41 | \( 1 + (-2.93 - 5.08i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.27 - 2.46i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.3 - 5.95i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 8.54iT - 53T^{2} \) |
| 59 | \( 1 + (0.525 + 0.910i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.58 - 7.94i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.50 + 2.02i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 2.02iT - 73T^{2} \) |
| 79 | \( 1 + (-3 + 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.49 + 2.59i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 + (0.764 + 0.441i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.788178788190977719040439216878, −8.978270210783893189901728136258, −7.82637419669022693034183368265, −6.97753963985417969248361722208, −6.27513916485018789379429474034, −5.87938945201590947873830822863, −4.38114170511558726360957749012, −3.46397232178542059988989432232, −2.75117870848288262383881885698, −1.24636225140191034768854084412,
0.55323881857158931347194670846, 2.32742816314195375237471773992, 2.89856675040354575802621745185, 4.25674126439953101302174762063, 5.38269643088027199401939487141, 5.79352741727436677575392987685, 6.82667426885668643469838519507, 7.46264648265336195971205491330, 8.819233083100707564038961331215, 9.378928627614823002274318702795