L(s) = 1 | + (−1.22 + 0.707i)2-s + (0.999 − 1.73i)4-s + (5.25 − 3.03i)5-s + 2.82i·8-s + (−4.29 + 7.43i)10-s + (10.5 + 6.06i)11-s + 18.5·13-s + (−2.00 − 3.46i)16-s + (−9.44 − 5.45i)17-s + (10 + 17.3i)19-s − 12.1i·20-s − 17.1·22-s + (−10.5 + 6.06i)23-s + (5.91 − 10.2i)25-s + (−22.7 + 13.1i)26-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (1.05 − 0.606i)5-s + 0.353i·8-s + (−0.429 + 0.743i)10-s + (0.955 + 0.551i)11-s + 1.42·13-s + (−0.125 − 0.216i)16-s + (−0.555 − 0.320i)17-s + (0.526 + 0.911i)19-s − 0.606i·20-s − 0.780·22-s + (−0.457 + 0.263i)23-s + (0.236 − 0.409i)25-s + (−0.875 + 0.505i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.940848880\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.940848880\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-5.25 + 3.03i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-10.5 - 6.06i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 18.5T + 169T^{2} \) |
| 17 | \( 1 + (9.44 + 5.45i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-10 - 17.3i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (10.5 - 6.06i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 41.8iT - 841T^{2} \) |
| 31 | \( 1 + (-12.5 + 21.7i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (19 + 32.9i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 60.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 83.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-14.6 + 8.48i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (81.4 + 47.0i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (50.4 + 29.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-7.83 - 13.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-66.3 + 114. i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 12.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (38.4 - 66.6i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (16.8 + 29.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 60.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-4.13 + 2.38i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 188.T + 9.40e3T^{2} \) |
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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
−9.616411288919367041946043958083, −9.279535550013544373602289654554, −8.485513603307890200183877413476, −7.50303093036779557467044216443, −6.38672369351720684700485393156, −5.91342172910428708435789203787, −4.84850129421103910859691393116, −3.62134305841585902634288915443, −1.91457550101544287865528097833, −1.14939723029524278417031192927,
0.982189537047271131950665002356, 2.14144054931658785290745086941, 3.22305854980543166554830758707, 4.30072062504350798494625686481, 5.98868135192814129898163880610, 6.28516530817176649382279244437, 7.32351143967010367854218037064, 8.560957533410972270921584746645, 9.040908723308059623118534434262, 9.905047450344223417945083238884