| L(s) = 1 | + (−8 + 13.8i)2-s + (94.3 − 163. i)3-s + (−127. − 221. i)4-s + (1.09e3 − 1.89e3i)5-s + (1.50e3 + 2.61e3i)6-s + 6.91e3·7-s + 4.09e3·8-s + (−7.96e3 − 1.37e4i)9-s + (1.74e4 + 3.02e4i)10-s + 3.78e4·11-s − 4.83e4·12-s + (−3.80e4 − 6.59e4i)13-s + (−5.53e4 + 9.58e4i)14-s + (−2.06e5 − 3.56e5i)15-s + (−3.27e4 + 5.67e4i)16-s + (−5.74e4 + 9.94e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.672 − 1.16i)3-s + (−0.249 − 0.433i)4-s + (0.781 − 1.35i)5-s + (0.475 + 0.823i)6-s + 1.08·7-s + 0.353·8-s + (−0.404 − 0.701i)9-s + (0.552 + 0.956i)10-s + 0.778·11-s − 0.672·12-s + (−0.369 − 0.640i)13-s + (−0.385 + 0.666i)14-s + (−1.05 − 1.82i)15-s + (−0.125 + 0.216i)16-s + (−0.166 + 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.88189 - 1.63623i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88189 - 1.63623i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8 - 13.8i)T \) |
| 19 | \( 1 + (4.16e4 - 5.66e5i)T \) |
| good | 3 | \( 1 + (-94.3 + 163. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.09e3 + 1.89e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 - 6.91e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.78e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (3.80e4 + 6.59e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (5.74e4 - 9.94e4i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 23 | \( 1 + (1.50e5 + 2.59e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (2.39e6 + 4.14e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 - 8.93e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.52e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (8.17e6 - 1.41e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (2.12e7 - 3.67e7i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (2.50e7 + 4.33e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-1.92e7 - 3.33e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (4.24e7 - 7.34e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (7.12e7 + 1.23e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-9.02e7 - 1.56e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (-7.16e7 + 1.24e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-3.77e7 + 6.53e7i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (9.93e7 - 1.72e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 1.57e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-5.16e8 - 8.94e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-2.76e8 + 4.79e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−13.95111284229758465179834174551, −13.10600204361806032999943704719, −11.96773602770520624499677569846, −9.829863579132279253797138332755, −8.459518482234266669272874717636, −7.959091110940390966529968027110, −6.24736517562225738688216046966, −4.80451545230329362171488841191, −1.85358240426827744917459289074, −1.04974480696015504278714285819,
1.95650453549481362196381123211, 3.23739012649902840792189032953, 4.70099850769408936075290369327, 6.92661760228147706458973903964, 8.745711404820815778347805958271, 9.733591903414116987003791354612, 10.68828118453467208754856389799, 11.63133914355532214797624745734, 13.87459686155520887150761013939, 14.44834449063535150497435140029
