| L(s) = 1 | + (59.3 − 19.2i)2-s + (−999. − 726. i)3-s + (−161. + 117. i)4-s + (875. − 2.69e3i)5-s + (−7.33e4 − 2.38e4i)6-s + (1.10e5 + 1.52e5i)7-s + (−1.57e5 + 2.16e5i)8-s + (3.07e5 + 9.45e5i)9-s − 1.76e5i·10-s + (−7.26e5 − 1.61e6i)11-s + 2.46e5·12-s + (−4.23e6 + 1.37e6i)13-s + (9.53e6 + 6.92e6i)14-s + (−2.83e6 + 2.05e6i)15-s + (−4.91e6 + 1.51e7i)16-s + (−1.01e7 − 3.29e6i)17-s + ⋯ |
| L(s) = 1 | + (0.927 − 0.301i)2-s + (−1.37 − 0.995i)3-s + (−0.0394 + 0.0286i)4-s + (0.0560 − 0.172i)5-s + (−1.57 − 0.510i)6-s + (0.943 + 1.29i)7-s + (−0.601 + 0.827i)8-s + (0.578 + 1.77i)9-s − 0.176i·10-s + (−0.410 − 0.912i)11-s + 0.0826·12-s + (−0.877 + 0.285i)13-s + (1.26 + 0.920i)14-s + (−0.248 + 0.180i)15-s + (−0.293 + 0.902i)16-s + (−0.420 − 0.136i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.417064 + 0.475366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.417064 + 0.475366i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (7.26e5 + 1.61e6i)T \) |
| good | 2 | \( 1 + (-59.3 + 19.2i)T + (3.31e3 - 2.40e3i)T^{2} \) |
| 3 | \( 1 + (999. + 726. i)T + (1.64e5 + 5.05e5i)T^{2} \) |
| 5 | \( 1 + (-875. + 2.69e3i)T + (-1.97e8 - 1.43e8i)T^{2} \) |
| 7 | \( 1 + (-1.10e5 - 1.52e5i)T + (-4.27e9 + 1.31e10i)T^{2} \) |
| 13 | \( 1 + (4.23e6 - 1.37e6i)T + (1.88e13 - 1.36e13i)T^{2} \) |
| 17 | \( 1 + (1.01e7 + 3.29e6i)T + (4.71e14 + 3.42e14i)T^{2} \) |
| 19 | \( 1 + (4.40e7 - 6.06e7i)T + (-6.83e14 - 2.10e15i)T^{2} \) |
| 23 | \( 1 + 6.86e7T + 2.19e16T^{2} \) |
| 29 | \( 1 + (2.80e8 + 3.86e8i)T + (-1.09e17 + 3.36e17i)T^{2} \) |
| 31 | \( 1 + (-3.10e7 - 9.56e7i)T + (-6.37e17 + 4.62e17i)T^{2} \) |
| 37 | \( 1 + (-1.04e9 + 7.62e8i)T + (2.03e18 - 6.26e18i)T^{2} \) |
| 41 | \( 1 + (2.00e9 - 2.75e9i)T + (-6.97e18 - 2.14e19i)T^{2} \) |
| 43 | \( 1 - 1.67e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + (1.34e10 + 9.76e9i)T + (3.59e19 + 1.10e20i)T^{2} \) |
| 53 | \( 1 + (4.22e9 + 1.29e10i)T + (-3.97e20 + 2.88e20i)T^{2} \) |
| 59 | \( 1 + (4.82e10 - 3.50e10i)T + (5.49e20 - 1.69e21i)T^{2} \) |
| 61 | \( 1 + (-8.54e10 - 2.77e10i)T + (2.14e21 + 1.56e21i)T^{2} \) |
| 67 | \( 1 - 6.26e10T + 8.18e21T^{2} \) |
| 71 | \( 1 + (-1.94e10 + 5.98e10i)T + (-1.32e22 - 9.64e21i)T^{2} \) |
| 73 | \( 1 + (2.25e10 + 3.10e10i)T + (-7.07e21 + 2.17e22i)T^{2} \) |
| 79 | \( 1 + (1.38e10 - 4.50e9i)T + (4.78e22 - 3.47e22i)T^{2} \) |
| 83 | \( 1 + (2.07e11 + 6.74e10i)T + (8.64e22 + 6.28e22i)T^{2} \) |
| 89 | \( 1 + 1.54e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + (-3.57e11 - 1.10e12i)T + (-5.61e23 + 4.07e23i)T^{2} \) |
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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
−17.96175291951018216574991462625, −16.76587454469336245920077472267, −14.68869598089722177752954492486, −13.06921958166058208793239664528, −12.09297061134047505987902215395, −11.29597201862842431731131114358, −8.270616486944785125849566409444, −5.95157470163578805650108542046, −4.99865755843214642082925258135, −2.06631946403796507871459909159,
0.25425451769041122788087089524, 4.37617712626388624984817594542, 4.97000420942152473708766259091, 6.82526408192228350583634837103, 9.990230242953727245694010402940, 11.04439424910682509313606822398, 12.74007327491690813938602025112, 14.49354464812270200827667862001, 15.52267264876324993172042342242, 17.08036809796914155046248129361
