| L(s) = 1 | + 1.04·3-s − 1.98·5-s − 2.61·7-s − 1.89·9-s − 6.22·11-s − 1.55·13-s − 2.08·15-s − 5.09·17-s − 2.93·19-s − 2.74·21-s + 0.880·23-s − 1.05·25-s − 5.14·27-s − 7.15·29-s + 2.19·31-s − 6.53·33-s + 5.19·35-s − 8.61·37-s − 1.63·39-s + 1.42·41-s − 4.95·43-s + 3.77·45-s + 4.40·47-s − 0.171·49-s − 5.34·51-s + 6.68·53-s + 12.3·55-s + ⋯ |
| L(s) = 1 | + 0.606·3-s − 0.888·5-s − 0.987·7-s − 0.632·9-s − 1.87·11-s − 0.431·13-s − 0.538·15-s − 1.23·17-s − 0.674·19-s − 0.598·21-s + 0.183·23-s − 0.210·25-s − 0.989·27-s − 1.32·29-s + 0.394·31-s − 1.13·33-s + 0.877·35-s − 1.41·37-s − 0.261·39-s + 0.222·41-s − 0.755·43-s + 0.562·45-s + 0.642·47-s − 0.0244·49-s − 0.748·51-s + 0.918·53-s + 1.66·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01567864936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01567864936\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 1.04T + 3T^{2} \) |
| 5 | \( 1 + 1.98T + 5T^{2} \) |
| 7 | \( 1 + 2.61T + 7T^{2} \) |
| 11 | \( 1 + 6.22T + 11T^{2} \) |
| 13 | \( 1 + 1.55T + 13T^{2} \) |
| 17 | \( 1 + 5.09T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 - 0.880T + 23T^{2} \) |
| 29 | \( 1 + 7.15T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 + 8.61T + 37T^{2} \) |
| 41 | \( 1 - 1.42T + 41T^{2} \) |
| 43 | \( 1 + 4.95T + 43T^{2} \) |
| 47 | \( 1 - 4.40T + 47T^{2} \) |
| 53 | \( 1 - 6.68T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 - 3.43T + 67T^{2} \) |
| 71 | \( 1 - 6.39T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 7.79T + 89T^{2} \) |
| 97 | \( 1 - 16.1T + 97T^{2} \) |
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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
−7.87914529593503189815617913383, −7.26110815098179504942734954564, −6.61480494060424570424868314775, −5.65055933609813800427244725675, −5.08409895094597304343395785775, −4.07421491097303578896073465197, −3.47273371113954048067080458679, −2.64253525276327421300713421948, −2.19272111851326216497333832154, −0.05378751101709106980216848045,
0.05378751101709106980216848045, 2.19272111851326216497333832154, 2.64253525276327421300713421948, 3.47273371113954048067080458679, 4.07421491097303578896073465197, 5.08409895094597304343395785775, 5.65055933609813800427244725675, 6.61480494060424570424868314775, 7.26110815098179504942734954564, 7.87914529593503189815617913383
