| L(s) = 1 | + (−1.07 + 0.913i)2-s + (1.40 − 3.00i)3-s + (0.330 − 1.97i)4-s + (−0.494 − 2.18i)5-s + (1.23 + 4.52i)6-s + (0.431 − 1.61i)7-s + (1.44 + 2.43i)8-s + (−5.12 − 6.11i)9-s + (2.52 + 1.90i)10-s + (3.75 + 2.16i)11-s + (−5.45 − 3.75i)12-s + (0.629 + 1.35i)13-s + (1.00 + 2.13i)14-s + (−7.24 − 1.56i)15-s + (−3.78 − 1.30i)16-s + (4.32 − 0.378i)17-s + ⋯ |
| L(s) = 1 | + (−0.763 + 0.645i)2-s + (0.808 − 1.73i)3-s + (0.165 − 0.986i)4-s + (−0.221 − 0.975i)5-s + (0.502 + 1.84i)6-s + (0.163 − 0.609i)7-s + (0.510 + 0.859i)8-s + (−1.70 − 2.03i)9-s + (0.798 + 0.601i)10-s + (1.13 + 0.653i)11-s + (−1.57 − 1.08i)12-s + (0.174 + 0.374i)13-s + (0.268 + 0.570i)14-s + (−1.86 − 0.404i)15-s + (−0.945 − 0.326i)16-s + (1.04 − 0.0917i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.653678 - 0.992156i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.653678 - 0.992156i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.07 - 0.913i)T \) |
| 5 | \( 1 + (0.494 + 2.18i)T \) |
| 19 | \( 1 + (2.54 - 3.54i)T \) |
| good | 3 | \( 1 + (-1.40 + 3.00i)T + (-1.92 - 2.29i)T^{2} \) |
| 7 | \( 1 + (-0.431 + 1.61i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.75 - 2.16i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.629 - 1.35i)T + (-8.35 + 9.95i)T^{2} \) |
| 17 | \( 1 + (-4.32 + 0.378i)T + (16.7 - 2.95i)T^{2} \) |
| 23 | \( 1 + (-2.23 - 1.56i)T + (7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (3.40 + 4.06i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (4.96 - 2.86i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.395 + 0.395i)T - 37iT^{2} \) |
| 41 | \( 1 + (-4.98 + 1.81i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.23 - 4.61i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (-5.38 - 0.471i)T + (46.2 + 8.16i)T^{2} \) |
| 53 | \( 1 + (-0.0428 + 0.0611i)T + (-18.1 - 49.8i)T^{2} \) |
| 59 | \( 1 + (2.70 + 2.27i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (2.32 + 13.1i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-10.6 - 0.933i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (-2.88 - 0.508i)T + (66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (3.30 + 1.54i)T + (46.9 + 55.9i)T^{2} \) |
| 79 | \( 1 + (8.70 - 3.16i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-9.12 - 2.44i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.716 - 1.96i)T + (-68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (8.89 - 0.777i)T + (95.5 - 16.8i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.21699846952721469464903322538, −9.555313083544710436496939027731, −9.030108196252172748111715764354, −8.029320543697515184054563622846, −7.53821993912104695064330350210, −6.68972860517198895032310353181, −5.69178513478680865296411493207, −3.95627635837101964386303873959, −1.83624302311020420903182416533, −1.02884999794887746376477885530,
2.51888592802322690591586174888, 3.39672323459557058084248439109, 4.13333486034858643769102431254, 5.72644679070827582002637566452, 7.32992754303284876333278870156, 8.465771547864175075487799771040, 9.031695686102398305023612808911, 9.770663113783895110290321936598, 10.80387993051160335745039643800, 11.03934223025147422213567563054
