| L(s) = 1 | + (2.77 + 15.7i)2-s + (−172. + 144. i)3-s + (−240. + 87.5i)4-s + (432. + 157. i)5-s + (−2.75e3 − 2.31e3i)6-s + (4.03e3 − 6.98e3i)7-s + (−2.04e3 − 3.54e3i)8-s + (5.35e3 − 3.03e4i)9-s + (−1.28e3 + 7.25e3i)10-s + (−1.18e4 − 2.04e4i)11-s + (2.87e4 − 4.98e4i)12-s + (−3.58e3 − 3.00e3i)13-s + (1.21e5 + 4.41e4i)14-s + (−9.72e4 + 3.54e4i)15-s + (5.02e4 − 4.21e4i)16-s + (7.70e3 + 4.36e4i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−1.22 + 1.02i)3-s + (−0.469 + 0.171i)4-s + (0.309 + 0.112i)5-s + (−0.867 − 0.728i)6-s + (0.635 − 1.10i)7-s + (−0.176 − 0.306i)8-s + (0.271 − 1.54i)9-s + (−0.0404 + 0.229i)10-s + (−0.243 − 0.421i)11-s + (0.400 − 0.693i)12-s + (−0.0347 − 0.0291i)13-s + (0.844 + 0.307i)14-s + (−0.496 + 0.180i)15-s + (0.191 − 0.160i)16-s + (0.0223 + 0.126i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.14543 + 0.369040i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.14543 + 0.369040i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.77 - 15.7i)T \) |
| 19 | \( 1 + (-5.14e5 - 2.41e5i)T \) |
| good | 3 | \( 1 + (172. - 144. i)T + (3.41e3 - 1.93e4i)T^{2} \) |
| 5 | \( 1 + (-432. - 157. i)T + (1.49e6 + 1.25e6i)T^{2} \) |
| 7 | \( 1 + (-4.03e3 + 6.98e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (1.18e4 + 2.04e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (3.58e3 + 3.00e3i)T + (1.84e9 + 1.04e10i)T^{2} \) |
| 17 | \( 1 + (-7.70e3 - 4.36e4i)T + (-1.11e11 + 4.05e10i)T^{2} \) |
| 23 | \( 1 + (-1.69e5 + 6.17e4i)T + (1.37e12 - 1.15e12i)T^{2} \) |
| 29 | \( 1 + (-1.85e5 + 1.05e6i)T + (-1.36e13 - 4.96e12i)T^{2} \) |
| 31 | \( 1 + (-2.47e4 + 4.28e4i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 2.13e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (9.10e6 - 7.64e6i)T + (5.68e13 - 3.22e14i)T^{2} \) |
| 43 | \( 1 + (-1.55e7 - 5.66e6i)T + (3.85e14 + 3.23e14i)T^{2} \) |
| 47 | \( 1 + (-8.20e6 + 4.65e7i)T + (-1.05e15 - 3.82e14i)T^{2} \) |
| 53 | \( 1 + (-3.66e7 + 1.33e7i)T + (2.52e15 - 2.12e15i)T^{2} \) |
| 59 | \( 1 + (2.81e7 + 1.59e8i)T + (-8.14e15 + 2.96e15i)T^{2} \) |
| 61 | \( 1 + (8.50e7 - 3.09e7i)T + (8.95e15 - 7.51e15i)T^{2} \) |
| 67 | \( 1 + (1.24e7 - 7.05e7i)T + (-2.55e16 - 9.30e15i)T^{2} \) |
| 71 | \( 1 + (-3.72e8 - 1.35e8i)T + (3.51e16 + 2.94e16i)T^{2} \) |
| 73 | \( 1 + (1.72e8 - 1.45e8i)T + (1.02e16 - 5.79e16i)T^{2} \) |
| 79 | \( 1 + (-5.14e8 + 4.31e8i)T + (2.08e16 - 1.18e17i)T^{2} \) |
| 83 | \( 1 + (-3.49e8 + 6.05e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (-2.66e7 - 2.23e7i)T + (6.08e16 + 3.45e17i)T^{2} \) |
| 97 | \( 1 + (-1.67e8 - 9.48e8i)T + (-7.14e17 + 2.60e17i)T^{2} \) |
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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
−14.59260122949013561991128508490, −13.47796362407278041173764416742, −11.73122040955374590002627967970, −10.65970070784770919491979749623, −9.715555936940798976756641119408, −7.80846781980274841294299826793, −6.22587092331998096981756456767, −5.09217673071242329380724816408, −3.96033538007911097938934680708, −0.63094912523932383819306167591,
1.06889442515943786672999239638, 2.30757852042991401136386903344, 5.01350951067490503577270334085, 5.93319942019844327263948180962, 7.61311588718360227382853336156, 9.336900862840320388461382941955, 11.00165999721461912119234301990, 11.85113901212720689774404115211, 12.60860178411503703671480822660, 13.72304898404817687958930526595
