L(s) = 1 | + (−1.62 − 0.593i)3-s + (1.50 + 2.60i)5-s + (−0.5 + 0.866i)7-s + (2.29 + 1.93i)9-s + (−2.74 + 4.75i)11-s + (−0.421 − 0.730i)13-s + (−0.901 − 5.13i)15-s − 4.59·17-s + 3.80·19-s + (1.32 − 1.11i)21-s + (−4.48 − 7.76i)23-s + (−2.02 + 3.51i)25-s + (−2.58 − 4.50i)27-s + (−0.974 + 1.68i)29-s + (1.68 + 2.92i)31-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)3-s + (0.672 + 1.16i)5-s + (−0.188 + 0.327i)7-s + (0.765 + 0.643i)9-s + (−0.826 + 1.43i)11-s + (−0.117 − 0.202i)13-s + (−0.232 − 1.32i)15-s − 1.11·17-s + 0.872·19-s + (0.289 − 0.242i)21-s + (−0.934 − 1.61i)23-s + (−0.405 + 0.702i)25-s + (−0.497 − 0.867i)27-s + (−0.180 + 0.313i)29-s + (0.303 + 0.525i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5592450149\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5592450149\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.62 + 0.593i)T \) |
| 7 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-1.50 - 2.60i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.74 - 4.75i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.421 + 0.730i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.59T + 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 + (4.48 + 7.76i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.974 - 1.68i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.68 - 2.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 + (3.69 + 6.39i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.11 + 5.39i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.57 - 11.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 8.05T + 53T^{2} \) |
| 59 | \( 1 + (-5.04 - 8.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.84 - 4.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.71 + 4.71i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.81T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + (-1.37 + 2.37i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.35 - 11.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 8.74T + 89T^{2} \) |
| 97 | \( 1 + (-3.83 + 6.64i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.31383864959767253033179842894, −9.966638739935855834286788597928, −8.699651265950965024277589599389, −7.39176777890900305593988588145, −6.95377831204204033351317592741, −6.16518804780135438532813436584, −5.29740127614698376960974896912, −4.38622024693116545185545997183, −2.70212005411369004476048397917, −1.97005257851253576978912194307,
0.27623428029690580170783483358, 1.60518462862470276107055490702, 3.38441544519663050975135861115, 4.50405303474622618483166463107, 5.43458629715013146553701841231, 5.80508719554487370836568844993, 6.88843904710009498708040750728, 8.040880672780514874079986013474, 8.914280976800630319816548607512, 9.715684206425772623200982132424