| L(s) = 1 | + (0.723 + 1.21i)2-s + (−1.52 − 0.822i)3-s + (−0.954 + 1.75i)4-s + (−0.0778 − 0.591i)5-s + (−0.102 − 2.44i)6-s + (0.748 + 2.79i)7-s + (−2.82 + 0.111i)8-s + (1.64 + 2.50i)9-s + (0.662 − 0.522i)10-s + (−1.99 + 1.53i)11-s + (2.90 − 1.89i)12-s + (−2.69 + 3.51i)13-s + (−2.85 + 2.93i)14-s + (−0.367 + 0.965i)15-s + (−2.17 − 3.35i)16-s + 1.23·17-s + ⋯ |
| L(s) = 1 | + (0.511 + 0.859i)2-s + (−0.880 − 0.474i)3-s + (−0.477 + 0.878i)4-s + (−0.0348 − 0.264i)5-s + (−0.0419 − 0.999i)6-s + (0.283 + 1.05i)7-s + (−0.999 + 0.0394i)8-s + (0.549 + 0.835i)9-s + (0.209 − 0.165i)10-s + (−0.602 + 0.462i)11-s + (0.837 − 0.546i)12-s + (−0.748 + 0.975i)13-s + (−0.763 + 0.783i)14-s + (−0.0949 + 0.249i)15-s + (−0.544 − 0.838i)16-s + 0.298·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.292168 + 0.881251i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.292168 + 0.881251i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.723 - 1.21i)T \) |
| 3 | \( 1 + (1.52 + 0.822i)T \) |
| good | 5 | \( 1 + (0.0778 + 0.591i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (-0.748 - 2.79i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (1.99 - 1.53i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (2.69 - 3.51i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 1.23T + 17T^{2} \) |
| 19 | \( 1 + (5.87 - 2.43i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.49 - 0.401i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-7.81 - 1.02i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (2.70 - 1.56i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.693 - 1.67i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-1.27 + 4.75i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-3.82 - 4.98i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-4.57 - 2.64i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.62 + 11.1i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (10.2 - 1.35i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (0.497 - 3.78i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-6.79 + 8.86i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (1.59 + 1.59i)T + 71iT^{2} \) |
| 73 | \( 1 + (9.90 - 9.90i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.21 + 2.09i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.85 - 1.29i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (9.32 - 9.32i)T - 89iT^{2} \) |
| 97 | \( 1 + (2.26 - 3.92i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.44907931525483171940576315212, −11.67765351500382939759295093623, −10.40513741961946559385326292783, −9.043308684197749710768520065373, −8.149879894185555615648135035789, −7.08896731705985592551891982162, −6.22333233358410399127193696218, −5.18751859286826755728992724529, −4.51117854350320797187000319088, −2.34972842844879693628007325953,
0.65661039258316493621441640380, 2.90365893923227478330593365023, 4.24152953856336710737515874804, 5.05153757197752377528384419927, 6.15275627605661778340917498418, 7.35820455984022900641033598225, 8.870709519290853134239663816629, 10.27491916430301835938680340557, 10.51607829216696797122001522134, 11.21050039254393356199894480212
