| L(s) = 1 | − 0.680·2-s − 1.53·4-s − 2.78·5-s − 0.191·7-s + 2.40·8-s + 1.89·10-s + 3.81·11-s + 0.780·13-s + 0.130·14-s + 1.43·16-s + 5.90·17-s + 4.26·19-s + 4.27·20-s − 2.59·22-s − 5.94·23-s + 2.73·25-s − 0.530·26-s + 0.293·28-s + 1.48·29-s − 7.36·31-s − 5.79·32-s − 4.01·34-s + 0.531·35-s + 6.39·37-s − 2.90·38-s − 6.69·40-s + 3.39·41-s + ⋯ |
| L(s) = 1 | − 0.480·2-s − 0.768·4-s − 1.24·5-s − 0.0722·7-s + 0.850·8-s + 0.598·10-s + 1.14·11-s + 0.216·13-s + 0.0347·14-s + 0.359·16-s + 1.43·17-s + 0.978·19-s + 0.956·20-s − 0.552·22-s − 1.24·23-s + 0.547·25-s − 0.104·26-s + 0.0555·28-s + 0.275·29-s − 1.32·31-s − 1.02·32-s − 0.688·34-s + 0.0899·35-s + 1.05·37-s − 0.470·38-s − 1.05·40-s + 0.529·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8957080912\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8957080912\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 + 0.680T + 2T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 7 | \( 1 + 0.191T + 7T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 13 | \( 1 - 0.780T + 13T^{2} \) |
| 17 | \( 1 - 5.90T + 17T^{2} \) |
| 19 | \( 1 - 4.26T + 19T^{2} \) |
| 23 | \( 1 + 5.94T + 23T^{2} \) |
| 29 | \( 1 - 1.48T + 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 - 6.39T + 37T^{2} \) |
| 41 | \( 1 - 3.39T + 41T^{2} \) |
| 43 | \( 1 + 7.56T + 43T^{2} \) |
| 47 | \( 1 + 4.15T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 6.15T + 61T^{2} \) |
| 67 | \( 1 + 1.97T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 - 1.74T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 4.97T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 7.76T + 97T^{2} \) |
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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
−8.395713189635509392399782461796, −7.73870108649873549038046690225, −7.39462928293062419138534395604, −6.27415287584278515006464220753, −5.41051684824778333396304168473, −4.51221717167619460544541716318, −3.74883571193518829688999159434, −3.36844389689149819416954251857, −1.58298003755834468443163867983, −0.62911691262229525999011178036,
0.62911691262229525999011178036, 1.58298003755834468443163867983, 3.36844389689149819416954251857, 3.74883571193518829688999159434, 4.51221717167619460544541716318, 5.41051684824778333396304168473, 6.27415287584278515006464220753, 7.39462928293062419138534395604, 7.73870108649873549038046690225, 8.395713189635509392399782461796
