| L(s) = 1 | + (−0.870 + 2.10i)2-s + (−1.40 − 0.936i)3-s + (−0.833 − 0.833i)4-s + (4.41 − 0.877i)5-s + (3.18 − 2.13i)6-s + (9.06 + 1.80i)7-s + (−5.93 + 2.45i)8-s + (−2.35 − 5.68i)9-s + (−1.99 + 10.0i)10-s + (−4.92 − 7.36i)11-s + (0.387 + 1.94i)12-s + (15.9 − 15.9i)13-s + (−11.6 + 17.4i)14-s + (−7.00 − 2.90i)15-s − 19.3i·16-s + ⋯ |
| L(s) = 1 | + (−0.435 + 1.05i)2-s + (−0.467 − 0.312i)3-s + (−0.208 − 0.208i)4-s + (0.882 − 0.175i)5-s + (0.531 − 0.355i)6-s + (1.29 + 0.257i)7-s + (−0.741 + 0.307i)8-s + (−0.261 − 0.632i)9-s + (−0.199 + 1.00i)10-s + (−0.447 − 0.669i)11-s + (0.0322 + 0.162i)12-s + (1.22 − 1.22i)13-s + (−0.834 + 1.24i)14-s + (−0.467 − 0.193i)15-s − 1.20i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.41320 + 0.372194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41320 + 0.372194i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (0.870 - 2.10i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (1.40 + 0.936i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (-4.41 + 0.877i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-9.06 - 1.80i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (4.92 + 7.36i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-15.9 + 15.9i)T - 169iT^{2} \) |
| 19 | \( 1 + (-2.25 + 5.44i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-20.0 + 13.4i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-5.05 - 25.4i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (-10.2 + 15.3i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (-44.6 - 29.8i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-15.5 - 3.09i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (-3.80 - 9.18i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (54.2 - 54.2i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-26.6 + 64.4i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (20.4 - 8.48i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-1.25 + 6.28i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 - 11.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (17.5 + 11.7i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-68.1 + 13.5i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-61.5 - 92.1i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (31.4 + 13.0i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-6.29 - 6.29i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (17.1 + 86.3i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
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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
−11.44642772427485032207129576006, −10.91963743322202267031136385298, −9.438684739416140392741967126553, −8.457374800891876781046806489622, −7.977714453080836126350675322010, −6.57927413390374627447286328603, −5.82652050183290696102891927352, −5.18117763766430804517313734602, −2.96020826700410039701559232548, −1.02101414102903067405690272255,
1.45979537827941427244862105267, 2.33030636616459390631430442587, 4.18698781413552030596315441905, 5.36227348779252583964750753683, 6.39140501624233380559767064968, 7.86414132461631991648092753808, 9.001523371366073865223853318920, 9.913817013814947455902438102416, 10.74576173382813394456110641910, 11.24225651037297684993626803935
