| L(s) = 1 | + (1.30 − 0.968i)2-s + (−1.04 + 1.38i)3-s + (0.180 − 0.601i)4-s + (−0.229 − 3.94i)5-s + (−0.0168 + 2.80i)6-s + (−2.15 + 1.52i)7-s + (0.760 + 2.08i)8-s + (−0.825 − 2.88i)9-s + (−4.12 − 4.91i)10-s + (3.26 − 4.96i)11-s + (0.644 + 0.876i)12-s + (−1.91 − 1.42i)13-s + (−1.32 + 4.07i)14-s + (5.69 + 3.79i)15-s + (4.06 + 2.67i)16-s + (−0.974 − 5.52i)17-s + ⋯ |
| L(s) = 1 | + (0.919 − 0.684i)2-s + (−0.601 + 0.798i)3-s + (0.0900 − 0.300i)4-s + (−0.102 − 1.76i)5-s + (−0.00686 + 1.14i)6-s + (−0.816 + 0.577i)7-s + (0.268 + 0.738i)8-s + (−0.275 − 0.961i)9-s + (−1.30 − 1.55i)10-s + (0.984 − 1.49i)11-s + (0.186 + 0.253i)12-s + (−0.531 − 0.395i)13-s + (−0.355 + 1.08i)14-s + (1.47 + 0.980i)15-s + (1.01 + 0.667i)16-s + (−0.236 − 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.808775 - 1.22804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.808775 - 1.22804i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.04 - 1.38i)T \) |
| 7 | \( 1 + (2.15 - 1.52i)T \) |
| good | 2 | \( 1 + (-1.30 + 0.968i)T + (0.573 - 1.91i)T^{2} \) |
| 5 | \( 1 + (0.229 + 3.94i)T + (-4.96 + 0.580i)T^{2} \) |
| 11 | \( 1 + (-3.26 + 4.96i)T + (-4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (1.91 + 1.42i)T + (3.72 + 12.4i)T^{2} \) |
| 17 | \( 1 + (0.974 + 5.52i)T + (-15.9 + 5.81i)T^{2} \) |
| 19 | \( 1 + (0.807 + 0.142i)T + (17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (0.935 + 3.94i)T + (-20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (0.106 + 0.0458i)T + (19.9 + 21.0i)T^{2} \) |
| 31 | \( 1 + (0.274 + 1.15i)T + (-27.7 + 13.9i)T^{2} \) |
| 37 | \( 1 + (0.391 - 0.328i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-1.73 - 0.202i)T + (39.8 + 9.45i)T^{2} \) |
| 43 | \( 1 + (-8.96 - 5.89i)T + (17.0 + 39.4i)T^{2} \) |
| 47 | \( 1 + (-2.17 - 0.516i)T + (42.0 + 21.0i)T^{2} \) |
| 53 | \( 1 - 4.33iT - 53T^{2} \) |
| 59 | \( 1 + (8.17 + 4.10i)T + (35.2 + 47.3i)T^{2} \) |
| 61 | \( 1 + (9.57 - 2.86i)T + (50.9 - 33.5i)T^{2} \) |
| 67 | \( 1 + (0.620 + 0.0725i)T + (65.1 + 15.4i)T^{2} \) |
| 71 | \( 1 + (-1.42 + 3.91i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-15.8 - 2.79i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-8.73 - 11.7i)T + (-22.6 + 75.6i)T^{2} \) |
| 83 | \( 1 + (-2.33 + 0.273i)T + (80.7 - 19.1i)T^{2} \) |
| 89 | \( 1 + (6.97 + 5.85i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-4.08 - 8.12i)T + (-57.9 + 77.8i)T^{2} \) |
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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
−10.79904536016281654960867265661, −9.369128035184556125385358580319, −9.112268425190671384504151605794, −8.127350327624083359960045149776, −6.23378279171752154329663914172, −5.48601978726219758777808208956, −4.72155795364553042363974969396, −3.91970380954640870113141221047, −2.84897842871204702782743121934, −0.67630781554499042907453497478,
1.97505617037737875373678235299, 3.56983014034294510411433565485, 4.43805771058970376758716482687, 5.98155572938158926943099800106, 6.47262472684169071355404437730, 7.16990701054966172693280140468, 7.48723713479578140620510696645, 9.561378660912410696855967339517, 10.31732237801873143516821334796, 10.99788451630638444511539750238
