L(s) = 1 | + (−11.5 − 19.9i)5-s + (−9.65e3 + 4.34e4i)7-s + (8.66e5 + 5.00e5i)11-s + 1.96e6i·13-s + (5.15e6 − 8.92e6i)17-s + (8.66e4 − 5.00e4i)19-s + (4.11e7 − 2.37e7i)23-s + (2.44e7 − 4.22e7i)25-s − 1.43e8i·29-s + (1.35e8 + 7.83e7i)31-s + (9.75e5 − 3.06e5i)35-s + (−8.38e7 − 1.45e8i)37-s + 1.37e7·41-s − 1.33e9·43-s + (7.44e8 + 1.28e9i)47-s + ⋯ |
L(s) = 1 | + (−0.00164 − 0.00285i)5-s + (−0.217 + 0.976i)7-s + (1.62 + 0.936i)11-s + 1.46i·13-s + (0.879 − 1.52i)17-s + (0.00802 − 0.00463i)19-s + (1.33 − 0.768i)23-s + (0.499 − 0.866i)25-s − 1.29i·29-s + (0.850 + 0.491i)31-s + (0.00313 − 0.000987i)35-s + (−0.198 − 0.344i)37-s + 0.0185·41-s − 1.38·43-s + (0.473 + 0.820i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.942 - 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(2.954083194\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.954083194\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (9.65e3 - 4.34e4i)T \) |
good | 5 | \( 1 + (11.5 + 19.9i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-8.66e5 - 5.00e5i)T + (1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.96e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (-5.15e6 + 8.92e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (-8.66e4 + 5.00e4i)T + (5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-4.11e7 + 2.37e7i)T + (4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.43e8iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (-1.35e8 - 7.83e7i)T + (1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (8.38e7 + 1.45e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 1.37e7T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.33e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-7.44e8 - 1.28e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (1.06e9 + 6.14e8i)T + (4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-2.83e9 + 4.90e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-8.76e9 + 5.06e9i)T + (2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (3.65e9 - 6.32e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 2.66e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (-1.27e10 - 7.36e9i)T + (1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-2.02e10 - 3.51e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 - 8.35e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-8.64e9 - 1.49e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 1.68e11iT - 7.15e21T^{2} \) |
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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
−9.768763640034941911995303063613, −9.353051936168043859197553187535, −8.446033139625912951257523894417, −6.91016260715169456977987777703, −6.53211327383314267472402436750, −5.06393806945463709100120520873, −4.23712067247353769674208378714, −2.88519728712918746607860976696, −1.89083366670463629008413299574, −0.74410151752741131669267086066,
0.883583748739807119348071767958, 1.27999649677189750713270771928, 3.35339646059269090390065761290, 3.60260497276324909764101028602, 5.14820440238852830966978035344, 6.18301463585664823924592373360, 7.10288097184905039718613851423, 8.160340849204175263282112977374, 9.053210668636217427324211411651, 10.21504100216213149648127054092