L(s) = 1 | − 2-s + (1.66 + 0.492i)3-s + 4-s + (1.80 − 1.04i)5-s + (−1.66 − 0.492i)6-s + (−0.800 − 2.52i)7-s − 8-s + (2.51 + 1.63i)9-s + (−1.80 + 1.04i)10-s + (1.07 + 1.86i)11-s + (1.66 + 0.492i)12-s + (0.217 − 3.59i)13-s + (0.800 + 2.52i)14-s + (3.50 − 0.839i)15-s + 16-s − 0.557·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.958 + 0.284i)3-s + 0.5·4-s + (0.806 − 0.465i)5-s + (−0.677 − 0.201i)6-s + (−0.302 − 0.953i)7-s − 0.353·8-s + (0.838 + 0.545i)9-s + (−0.570 + 0.329i)10-s + (0.325 + 0.563i)11-s + (0.479 + 0.142i)12-s + (0.0602 − 0.998i)13-s + (0.213 + 0.673i)14-s + (0.905 − 0.216i)15-s + 0.250·16-s − 0.135·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61737 - 0.349310i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61737 - 0.349310i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-1.66 - 0.492i)T \) |
| 7 | \( 1 + (0.800 + 2.52i)T \) |
| 13 | \( 1 + (-0.217 + 3.59i)T \) |
good | 5 | \( 1 + (-1.80 + 1.04i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.07 - 1.86i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 0.557T + 17T^{2} \) |
| 19 | \( 1 + (-1.94 + 3.37i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.07iT - 23T^{2} \) |
| 29 | \( 1 + (6.10 + 3.52i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.21 + 5.57i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.31iT - 37T^{2} \) |
| 41 | \( 1 + (0.532 + 0.307i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.33 - 7.50i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.507 - 0.292i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.68 - 3.85i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 1.56iT - 59T^{2} \) |
| 61 | \( 1 + (7.10 + 4.10i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.3 + 7.12i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.52 + 11.3i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.198 + 0.344i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.73 - 9.93i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 8.70iT - 89T^{2} \) |
| 97 | \( 1 + (-1.64 - 2.85i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.37253285472658954182599442490, −9.548359869846615480563252600223, −9.377478020855768359483160717589, −8.046504741147019201282426733430, −7.48802644802021479647214463808, −6.38799545445637185945230381833, −5.05942592056548311979833869873, −3.84124949914012967641198420177, −2.60535815398715891422617382045, −1.25436041169096656686327910004,
1.72847064197837172231619726154, 2.60154074181411291198184108226, 3.75908108901051762859949468722, 5.64664683185002825563215018613, 6.50516685126834858532108523390, 7.29422398389410091950465051154, 8.542811991901158786245348297482, 9.016252164949866405370246158498, 9.729787371128267003325091549939, 10.57395268529227303131831203085