L(s) = 1 | + (−0.551 + 0.955i)2-s + (0.391 + 0.678i)4-s − 0.105·5-s − 3.07·8-s + (0.0581 − 0.100i)10-s − 3.33·11-s + (−1.23 + 2.14i)13-s + (0.909 − 1.57i)16-s + (0.806 − 1.39i)17-s + (3.84 + 6.65i)19-s + (−0.0413 − 0.0715i)20-s + (1.84 − 3.18i)22-s + 1.89·23-s − 4.98·25-s + (−1.36 − 2.36i)26-s + ⋯ |
L(s) = 1 | + (−0.389 + 0.675i)2-s + (0.195 + 0.339i)4-s − 0.0471·5-s − 1.08·8-s + (0.0183 − 0.0318i)10-s − 1.00·11-s + (−0.343 + 0.595i)13-s + (0.227 − 0.393i)16-s + (0.195 − 0.338i)17-s + (0.881 + 1.52i)19-s + (−0.00924 − 0.0160i)20-s + (0.392 − 0.679i)22-s + 0.395·23-s − 0.997·25-s + (−0.268 − 0.464i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2169469162\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2169469162\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.551 - 0.955i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 0.105T + 5T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 13 | \( 1 + (1.23 - 2.14i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.806 + 1.39i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.84 - 6.65i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 1.89T + 23T^{2} \) |
| 29 | \( 1 + (4.64 + 8.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.63 + 8.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.991 - 1.71i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.74 - 6.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.77 + 6.53i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.59 - 2.76i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.98 - 8.64i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.22 - 3.86i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.83 + 4.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.98 + 8.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 + (-2.36 + 4.09i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.84 - 6.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.584 - 1.01i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.01 - 5.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.90 + 3.29i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.695962244983025703422797340946, −9.533166019645848971655621464247, −8.155088867540074957261911654691, −7.82437124476764411569209317072, −7.14970326689605125692769906792, −6.03817278273476803894839552014, −5.48501790445009436441059429845, −4.14877612853405647654250909858, −3.15077531757250844360015611363, −2.03020935017676620851211511502,
0.095412219539733853448087797544, 1.54228321320418329936668592982, 2.71720449862479353619081792612, 3.44302416439731652614092546794, 5.16346608033450223952223059221, 5.42659043847161901039087381259, 6.77256155852632955707808819680, 7.44308773693445104881724109925, 8.507841197789271820852411601301, 9.254458883751526717609222155225