| L(s) = 1 | + 1.50i·5-s + 0.0279i·7-s − 1.76i·11-s + (−2.93 − 2.09i)13-s + 5.53·17-s + 4.53i·19-s − 0.748·23-s + 2.73·25-s − 4.23·29-s + 5.57i·31-s − 0.0420·35-s + 5.05i·37-s − 7.25i·41-s + 9.25·43-s − 2.02i·47-s + ⋯ |
| L(s) = 1 | + 0.672i·5-s + 0.0105i·7-s − 0.531i·11-s + (−0.812 − 0.582i)13-s + 1.34·17-s + 1.04i·19-s − 0.156·23-s + 0.547·25-s − 0.786·29-s + 1.00i·31-s − 0.00711·35-s + 0.831i·37-s − 1.13i·41-s + 1.41·43-s − 0.295i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 - 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.682317988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.682317988\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (2.93 + 2.09i)T \) |
| good | 5 | \( 1 - 1.50iT - 5T^{2} \) |
| 7 | \( 1 - 0.0279iT - 7T^{2} \) |
| 11 | \( 1 + 1.76iT - 11T^{2} \) |
| 17 | \( 1 - 5.53T + 17T^{2} \) |
| 19 | \( 1 - 4.53iT - 19T^{2} \) |
| 23 | \( 1 + 0.748T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 - 5.57iT - 31T^{2} \) |
| 37 | \( 1 - 5.05iT - 37T^{2} \) |
| 41 | \( 1 + 7.25iT - 41T^{2} \) |
| 43 | \( 1 - 9.25T + 43T^{2} \) |
| 47 | \( 1 + 2.02iT - 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 3.60iT - 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 - 7.85iT - 67T^{2} \) |
| 71 | \( 1 - 12.8iT - 71T^{2} \) |
| 73 | \( 1 - 2.35iT - 73T^{2} \) |
| 79 | \( 1 - 9.07T + 79T^{2} \) |
| 83 | \( 1 - 14.2iT - 83T^{2} \) |
| 89 | \( 1 + 1.13iT - 89T^{2} \) |
| 97 | \( 1 - 9.83iT - 97T^{2} \) |
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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
−8.872212408539946511153133167421, −8.041549523523086796138750868028, −7.43735321140151211251336714769, −6.72689427499155140491989134317, −5.68123329806550319516473477144, −5.31374288979399075560107631185, −3.97406969723482733230435522138, −3.24900005836425689867537848289, −2.41562680468080088541139570463, −1.03246888576653474825497091250,
0.65087535150632970694177558664, 1.90823320050527168912442086781, 2.89558073773480273769640241863, 4.08936968456809481157460846294, 4.76259075488157409330781802532, 5.49416534749140993442192281667, 6.37064814551255556207573573861, 7.48332805663585008262797984387, 7.64629627078919202288689870047, 8.923100323412995050091268858512
