| L(s) = 1 | + (−2.93 + 2.13i)2-s + (1.72 + 0.181i)3-s + (2.82 − 8.70i)4-s + (−1.33 + 2.31i)5-s + (−5.44 + 3.14i)6-s + (−11.6 + 2.47i)7-s + (5.77 + 17.7i)8-s + (2.93 + 0.623i)9-s + (−1.01 − 9.65i)10-s + (−10.2 + 9.20i)11-s + (6.44 − 14.4i)12-s + (−6.79 − 15.2i)13-s + (28.8 − 32.0i)14-s + (−2.72 + 3.74i)15-s + (−25.2 − 18.3i)16-s + (−11.1 − 10.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.46 + 1.06i)2-s + (0.574 + 0.0603i)3-s + (0.707 − 2.17i)4-s + (−0.267 + 0.463i)5-s + (−0.906 + 0.523i)6-s + (−1.66 + 0.352i)7-s + (0.722 + 2.22i)8-s + (0.326 + 0.0693i)9-s + (−0.101 − 0.965i)10-s + (−0.929 + 0.836i)11-s + (0.537 − 1.20i)12-s + (−0.522 − 1.17i)13-s + (2.06 − 2.28i)14-s + (−0.181 + 0.249i)15-s + (−1.57 − 1.14i)16-s + (−0.654 − 0.589i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0643143 - 0.196258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0643143 - 0.196258i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.72 - 0.181i)T \) |
| 31 | \( 1 + (-18.4 - 24.9i)T \) |
| good | 2 | \( 1 + (2.93 - 2.13i)T + (1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (1.33 - 2.31i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (11.6 - 2.47i)T + (44.7 - 19.9i)T^{2} \) |
| 11 | \( 1 + (10.2 - 9.20i)T + (12.6 - 120. i)T^{2} \) |
| 13 | \( 1 + (6.79 + 15.2i)T + (-113. + 125. i)T^{2} \) |
| 17 | \( 1 + (11.1 + 10.0i)T + (30.2 + 287. i)T^{2} \) |
| 19 | \( 1 + (-13.7 - 6.13i)T + (241. + 268. i)T^{2} \) |
| 23 | \( 1 + (24.3 - 7.90i)T + (427. - 310. i)T^{2} \) |
| 29 | \( 1 + (3.42 + 4.70i)T + (-259. + 799. i)T^{2} \) |
| 37 | \( 1 + (41.8 - 24.1i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (1.96 + 18.6i)T + (-1.64e3 + 349. i)T^{2} \) |
| 43 | \( 1 + (-4.85 + 10.8i)T + (-1.23e3 - 1.37e3i)T^{2} \) |
| 47 | \( 1 + (-51.4 - 37.3i)T + (682. + 2.10e3i)T^{2} \) |
| 53 | \( 1 + (-0.944 + 4.44i)T + (-2.56e3 - 1.14e3i)T^{2} \) |
| 59 | \( 1 + (5.61 - 53.3i)T + (-3.40e3 - 723. i)T^{2} \) |
| 61 | \( 1 - 116. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (-27.2 + 47.2i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (21.1 + 4.49i)T + (4.60e3 + 2.05e3i)T^{2} \) |
| 73 | \( 1 + (39.4 - 35.5i)T + (557. - 5.29e3i)T^{2} \) |
| 79 | \( 1 + (38.4 + 34.6i)T + (652. + 6.20e3i)T^{2} \) |
| 83 | \( 1 + (-57.4 + 6.03i)T + (6.73e3 - 1.43e3i)T^{2} \) |
| 89 | \( 1 + (35.0 + 11.3i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-32.8 + 101. i)T + (-7.61e3 - 5.53e3i)T^{2} \) |
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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
−15.07963489449922304870697850785, −13.63409819040148018104492779363, −12.35789566088022278316141893936, −10.35957140309678165067355211302, −9.928054402623490490236825835702, −8.911723554221494402645713690663, −7.62359732375504110467162987897, −6.97649699924521751740851745467, −5.59176215044244952083323057723, −2.84553555674819987260747224231,
0.22351928139857732785187601625, 2.50225243936854761543371972318, 3.80212857877455166781316750362, 6.71816219846383439046467047164, 7.987709254202647069759231571489, 8.980058175835742717932057727037, 9.760408834029377929060318934276, 10.66407756937255931887779781310, 11.99122001146826350197449134366, 12.84421838713443905482241301301
