L(s) = 1 | + (0.705 − 2.63i)2-s + i·3-s + (−4.70 − 2.71i)4-s + (−0.914 − 3.41i)5-s + (2.63 + 0.705i)6-s + (2.62 + 0.362i)7-s + (−6.62 + 6.62i)8-s − 9-s − 9.64·10-s + (0.312 − 0.312i)11-s + (2.71 − 4.70i)12-s + (−3.60 − 0.0874i)13-s + (2.80 − 6.64i)14-s + (3.41 − 0.914i)15-s + (7.34 + 12.7i)16-s + (0.715 − 1.23i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 1.86i)2-s + 0.577i·3-s + (−2.35 − 1.35i)4-s + (−0.409 − 1.52i)5-s + (1.07 + 0.288i)6-s + (0.990 + 0.137i)7-s + (−2.34 + 2.34i)8-s − 0.333·9-s − 3.04·10-s + (0.0942 − 0.0942i)11-s + (0.784 − 1.35i)12-s + (−0.999 − 0.0242i)13-s + (0.749 − 1.77i)14-s + (0.881 − 0.236i)15-s + (1.83 + 3.18i)16-s + (0.173 − 0.300i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.164482 + 1.24503i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.164482 + 1.24503i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (-2.62 - 0.362i)T \) |
| 13 | \( 1 + (3.60 + 0.0874i)T \) |
good | 2 | \( 1 + (-0.705 + 2.63i)T + (-1.73 - i)T^{2} \) |
| 5 | \( 1 + (0.914 + 3.41i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.312 + 0.312i)T - 11iT^{2} \) |
| 17 | \( 1 + (-0.715 + 1.23i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.931 - 0.931i)T - 19iT^{2} \) |
| 23 | \( 1 + (-6.22 + 3.59i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.82 + 6.61i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.698 - 0.187i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (0.982 + 0.263i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.748 - 2.79i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-7.20 + 4.15i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.906 + 0.242i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.48 - 4.30i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.14 - 0.841i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + 13.0iT - 61T^{2} \) |
| 67 | \( 1 + (-4.49 - 4.49i)T + 67iT^{2} \) |
| 71 | \( 1 + (-0.487 + 1.82i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (3.29 - 12.2i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.77 - 3.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.33 - 2.33i)T - 83iT^{2} \) |
| 89 | \( 1 + (2.70 - 10.1i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.51 - 0.673i)T + (84.0 + 48.5i)T^{2} \) |
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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
−11.55462635094217481189100046591, −10.68232163143246162637407994293, −9.645844230729352991415112894794, −8.913011257597270707499733217567, −8.126616110993377934562272224440, −5.40629049486089370846454104546, −4.75736504774665894631592147619, −4.13615462468224721690386811604, −2.49431517496025673433164248094, −0.897320066369147986918372616639,
3.09044683216902559785865209783, 4.50930222074361468743310429862, 5.61025164698421465405126459544, 6.84838345048811520837057127857, 7.24329919954545697643433949927, 7.965031248703986724136424307074, 9.059211949655963261178534311651, 10.55237164977011500170052485687, 11.70074832119352106642262761283, 12.69857504291248207393988055131