L(s) = 1 | − 2.33·2-s + (1.15 − 1.99i)3-s + 3.43·4-s + (1.68 − 2.91i)5-s + (−2.69 + 4.66i)6-s − 3.34·8-s + (−1.16 − 2.01i)9-s + (−3.92 + 6.80i)10-s + (−1.16 + 2.01i)11-s + (3.96 − 6.87i)12-s + (−0.408 − 3.58i)13-s + (−3.89 − 6.74i)15-s + 0.933·16-s + 5.45·17-s + (2.71 + 4.70i)18-s + (−3.58 − 6.20i)19-s + ⋯ |
L(s) = 1 | − 1.64·2-s + (0.666 − 1.15i)3-s + 1.71·4-s + (0.753 − 1.30i)5-s + (−1.09 + 1.90i)6-s − 1.18·8-s + (−0.388 − 0.673i)9-s + (−1.24 + 2.15i)10-s + (−0.351 + 0.608i)11-s + (1.14 − 1.98i)12-s + (−0.113 − 0.993i)13-s + (−1.00 − 1.74i)15-s + 0.233·16-s + 1.32·17-s + (0.640 + 1.10i)18-s + (−0.822 − 1.42i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.252451 - 0.844441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.252451 - 0.844441i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (0.408 + 3.58i)T \) |
good | 2 | \( 1 + 2.33T + 2T^{2} \) |
| 3 | \( 1 + (-1.15 + 1.99i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.68 + 2.91i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.16 - 2.01i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 - 5.45T + 17T^{2} \) |
| 19 | \( 1 + (3.58 + 6.20i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 6.45T + 23T^{2} \) |
| 29 | \( 1 + (-4.22 - 7.31i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.52 + 2.64i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.05T + 37T^{2} \) |
| 41 | \( 1 + (0.468 + 0.812i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.04 + 3.54i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.73 - 2.99i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.17 - 2.02i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 7.24T + 59T^{2} \) |
| 61 | \( 1 + (-3.19 - 5.53i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.30 - 3.99i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.79 + 6.57i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.03 + 1.79i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.79 + 6.57i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 2.89T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + (1.77 - 3.08i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.959248288946991702269254996789, −9.141913273120150323645932037999, −8.481226904589051868692541887940, −7.85889507484750269714745716379, −7.18909228159948037538798500749, −6.04599615811058381188619099883, −4.86142367958467943862486740856, −2.65695754868378917317304619241, −1.72888373508026676429052858210, −0.76423655163299994550154811930,
1.93736358616088475201393040636, 2.96319370209523764930603837497, 4.07960432695075579383628748896, 5.89271212821711254313855585216, 6.64531929480115688973155720272, 7.88309219720193776464377792566, 8.397817216332858537120344564562, 9.588823870713672217835834768173, 9.903321990126708379427809311791, 10.40674757555282032997452833836