L(s) = 1 | + (−0.169 + 1.99i)2-s + (−3.94 − 0.675i)4-s + 12.3·7-s + (2.01 − 7.74i)8-s + 11.0i·11-s − 2.82i·13-s + (−2.10 + 24.7i)14-s + (15.0 + 5.32i)16-s − 6.52i·17-s − 27.9i·19-s + (−22.0 − 1.87i)22-s − 7.90·23-s + (5.61 + 0.477i)26-s + (−48.8 − 8.37i)28-s + 50.7·29-s + ⋯ |
L(s) = 1 | + (−0.0847 + 0.996i)2-s + (−0.985 − 0.168i)4-s + 1.77·7-s + (0.251 − 0.967i)8-s + 1.00i·11-s − 0.216i·13-s + (−0.150 + 1.76i)14-s + (0.942 + 0.332i)16-s − 0.383i·17-s − 1.47i·19-s + (−1.00 − 0.0850i)22-s − 0.343·23-s + (0.216 + 0.0183i)26-s + (−1.74 − 0.298i)28-s + 1.74·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.591 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.027720095\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.027720095\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.169 - 1.99i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 12.3T + 49T^{2} \) |
| 11 | \( 1 - 11.0iT - 121T^{2} \) |
| 13 | \( 1 + 2.82iT - 169T^{2} \) |
| 17 | \( 1 + 6.52iT - 289T^{2} \) |
| 19 | \( 1 + 27.9iT - 361T^{2} \) |
| 23 | \( 1 + 7.90T + 529T^{2} \) |
| 29 | \( 1 - 50.7T + 841T^{2} \) |
| 31 | \( 1 + 36.3iT - 961T^{2} \) |
| 37 | \( 1 + 18.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 5.30T + 1.68e3T^{2} \) |
| 43 | \( 1 - 45.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 11.7T + 2.20e3T^{2} \) |
| 53 | \( 1 - 41.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 10.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 56.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 16.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 66.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 15.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 123. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 99.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 101.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 127. iT - 9.40e3T^{2} \) |
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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
−9.857918195328164874114586282559, −8.976196818485410999940255246121, −8.198713316166618130311710008914, −7.52438473870303379240285600377, −6.81402735964587291102406366344, −5.59396656842930350506439835917, −4.75516375114870954499488033362, −4.29181697142428871542522567822, −2.37677219822765580275356721615, −0.915022416472905920345129770646,
1.06980260750143217135890745169, 1.95352804442867363637546423225, 3.28844718317283130865224173782, 4.33383984608980915694063368371, 5.12056225122096555097192508822, 6.08709353285073408452420057039, 7.66684945166187840086105728680, 8.387305861122875945109920895345, 8.745555722083993367315964121654, 10.14793013822459239186667240723