| L(s) = 1 | + (−1.13 − 0.367i)2-s + (1.09 − 1.34i)3-s + (−0.473 − 0.343i)4-s + (−3.25 − 1.88i)5-s + (−1.73 + 1.11i)6-s + (−0.121 + 1.15i)7-s + (1.80 + 2.48i)8-s + (−0.608 − 2.93i)9-s + (2.99 + 3.32i)10-s + (1.44 − 0.642i)11-s + (−0.979 + 0.259i)12-s + (1.12 − 5.27i)13-s + (0.561 − 1.26i)14-s + (−6.09 + 2.31i)15-s + (−0.768 − 2.36i)16-s + (3.70 + 1.64i)17-s + ⋯ |
| L(s) = 1 | + (−0.799 − 0.259i)2-s + (0.631 − 0.775i)3-s + (−0.236 − 0.171i)4-s + (−1.45 − 0.841i)5-s + (−0.706 + 0.456i)6-s + (−0.0458 + 0.436i)7-s + (0.639 + 0.879i)8-s + (−0.202 − 0.979i)9-s + (0.947 + 1.05i)10-s + (0.434 − 0.193i)11-s + (−0.282 + 0.0749i)12-s + (0.310 − 1.46i)13-s + (0.150 − 0.336i)14-s + (−1.57 + 0.598i)15-s + (−0.192 − 0.591i)16-s + (0.898 + 0.400i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.550 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.277836 - 0.515859i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.277836 - 0.515859i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.09 + 1.34i)T \) |
| 31 | \( 1 + (-1.59 - 5.33i)T \) |
| good | 2 | \( 1 + (1.13 + 0.367i)T + (1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (3.25 + 1.88i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (0.121 - 1.15i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-1.44 + 0.642i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-1.12 + 5.27i)T + (-11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-3.70 - 1.64i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-2.07 + 0.440i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.05 + 0.765i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.285 - 0.878i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (7.98 - 4.60i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.74 - 2.47i)T + (4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (1.64 + 7.73i)T + (-39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (-7.19 + 2.33i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.927 + 8.82i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-7.74 - 6.97i)T + (6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 - 4.86iT - 61T^{2} \) |
| 67 | \( 1 + (-1.87 + 3.25i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.89 - 0.830i)T + (69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-0.894 - 2.00i)T + (-48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-1.85 + 4.16i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-3.43 - 3.81i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (2.01 + 1.46i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-8.65 - 6.28i)T + (29.9 + 92.2i)T^{2} \) |
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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
−13.55457938009503971533680992581, −12.43768009877417459060766434249, −11.75424632823819637172685186970, −10.29041201143540647343117264119, −8.756615265536735531600210487053, −8.421903403971530080864058213625, −7.42592255562039999184324695007, −5.34531636373579557990185832437, −3.48230388842460091224325232691, −0.964355086149515925092115439201,
3.54723641041092790363728744340, 4.31141460811670642842572068557, 7.07165030109432643998981550701, 7.76103339275406441976070750242, 8.889000548619845615207682687388, 9.857776390800153569184270146835, 10.98292701687059256848375817791, 11.99286216308530182055330578726, 13.75360645236177219939084614478, 14.49857668842356827996976561570
