| L(s) = 1 | + (−0.893 + 0.448i)2-s + (0.647 + 1.60i)3-s + (0.597 − 0.802i)4-s + (−0.750 + 2.50i)5-s + (−1.29 − 1.14i)6-s + (0.318 − 0.738i)7-s + (−0.173 + 0.984i)8-s + (−2.16 + 2.08i)9-s + (−0.454 − 2.57i)10-s + (0.151 + 0.0358i)11-s + (1.67 + 0.439i)12-s + (−0.574 + 0.377i)13-s + (0.0467 + 0.802i)14-s + (−4.51 + 0.417i)15-s + (−0.286 − 0.957i)16-s + (0.0626 + 0.0228i)17-s + ⋯ |
| L(s) = 1 | + (−0.631 + 0.317i)2-s + (0.373 + 0.927i)3-s + (0.298 − 0.401i)4-s + (−0.335 + 1.12i)5-s + (−0.530 − 0.467i)6-s + (0.120 − 0.279i)7-s + (−0.0613 + 0.348i)8-s + (−0.720 + 0.693i)9-s + (−0.143 − 0.814i)10-s + (0.0456 + 0.0108i)11-s + (0.483 + 0.126i)12-s + (−0.159 + 0.104i)13-s + (0.0124 + 0.214i)14-s + (−1.16 + 0.107i)15-s + (−0.0717 − 0.239i)16-s + (0.0152 + 0.00553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.456432 + 0.740584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.456432 + 0.740584i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.893 - 0.448i)T \) |
| 3 | \( 1 + (-0.647 - 1.60i)T \) |
| good | 5 | \( 1 + (0.750 - 2.50i)T + (-4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (-0.318 + 0.738i)T + (-4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (-0.151 - 0.0358i)T + (9.82 + 4.93i)T^{2} \) |
| 13 | \( 1 + (0.574 - 0.377i)T + (5.14 - 11.9i)T^{2} \) |
| 17 | \( 1 + (-0.0626 - 0.0228i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (0.221 - 0.0805i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-3.45 - 8.00i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (-0.585 + 10.0i)T + (-28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (-7.28 - 0.851i)T + (30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (-6.12 + 5.14i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (4.49 + 2.25i)T + (24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (-2.28 + 2.42i)T + (-2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-12.4 + 1.46i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (3.27 + 5.66i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7.05 - 1.67i)T + (52.7 - 26.4i)T^{2} \) |
| 61 | \( 1 + (3.70 + 4.97i)T + (-17.4 + 58.4i)T^{2} \) |
| 67 | \( 1 + (-0.484 - 8.31i)T + (-66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (0.683 + 3.87i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.74 - 9.87i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (15.0 - 7.56i)T + (47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (-1.94 + 0.978i)T + (49.5 - 66.5i)T^{2} \) |
| 89 | \( 1 + (0.587 - 3.33i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.07 - 6.93i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−13.61043327753766998860667653022, −11.68977113886325378238178599289, −10.95866551637792152243733584448, −10.10665075342818736966915759419, −9.262146781904419212112159421894, −7.998875861587111387639764035229, −7.14736609278065272093537727953, −5.74509898166297266145120865565, −4.13436773128561259978791859694, −2.74361528887922098859392503470,
1.06541587930599642142530118216, 2.79919556725813851265055859874, 4.70421315745547264587977367909, 6.36815678476263204201859861054, 7.60147306876240058402267158227, 8.594942625521222645694712679662, 9.029475269607285735182311769267, 10.54747531944280915082387666334, 11.84865705491358569538257158075, 12.44121825097002199239129941929
