| L(s) = 1 | + (0.429 − 1.34i)2-s + (−1.65 + 0.237i)3-s + (−1.63 − 1.15i)4-s + (0.934 − 3.18i)5-s + (−0.388 + 2.32i)6-s + (0.148 − 0.171i)7-s + (−2.25 + 1.70i)8-s + (−0.208 + 0.0611i)9-s + (−3.88 − 2.62i)10-s + (3.48 − 2.24i)11-s + (2.96 + 1.52i)12-s + (3.81 + 4.40i)13-s + (−0.166 − 0.273i)14-s + (−0.787 + 5.47i)15-s + (1.32 + 3.77i)16-s + (−0.449 − 0.205i)17-s + ⋯ |
| L(s) = 1 | + (0.303 − 0.952i)2-s + (−0.953 + 0.137i)3-s + (−0.815 − 0.578i)4-s + (0.417 − 1.42i)5-s + (−0.158 + 0.949i)6-s + (0.0560 − 0.0646i)7-s + (−0.798 + 0.601i)8-s + (−0.0694 + 0.0203i)9-s + (−1.22 − 0.830i)10-s + (1.05 − 0.675i)11-s + (0.856 + 0.439i)12-s + (1.05 + 1.22i)13-s + (−0.0446 − 0.0730i)14-s + (−0.203 + 1.41i)15-s + (0.330 + 0.943i)16-s + (−0.109 − 0.0497i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.449395 - 0.725247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.449395 - 0.725247i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.429 + 1.34i)T \) |
| 23 | \( 1 + (-4.22 - 2.26i)T \) |
| good | 3 | \( 1 + (1.65 - 0.237i)T + (2.87 - 0.845i)T^{2} \) |
| 5 | \( 1 + (-0.934 + 3.18i)T + (-4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.148 + 0.171i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-3.48 + 2.24i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-3.81 - 4.40i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.449 + 0.205i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.608 + 1.33i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (0.299 - 0.654i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (9.23 + 1.32i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (1.33 + 4.54i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (2.79 + 0.819i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.508 - 3.53i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 - 6.02iT - 47T^{2} \) |
| 53 | \( 1 + (-4.56 - 3.95i)T + (7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.34 + 3.76i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (9.72 + 1.39i)T + (58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-10.2 - 6.57i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (2.54 - 3.96i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (2.85 + 6.24i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.34 - 1.55i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (6.97 - 2.04i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-14.1 + 2.04i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (0.265 - 0.903i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−13.46837085245314303597248977229, −12.51747905137868452378272505797, −11.50014100105130410392549961564, −10.97042545873490686150753235149, −9.208947896974478949524522697385, −8.856064713518060557571846446522, −6.20128466316384754853019725181, −5.22985643090105605183316829590, −4.04876483055653145324142287308, −1.28684954387240792460417384019,
3.47560420964613575475439380992, 5.42616095019439225766954886458, 6.38536363769099656017042662978, 7.08818246911051218444100628981, 8.708352811159993786213881231719, 10.22784321145415876708453635739, 11.21984668486212475075050706765, 12.39292460356516442225632960705, 13.53671118855490798109044251362, 14.69810293160715286639470130198
