| L(s) = 1 | − 2.54·2-s − 3-s + 4.47·4-s + 0.653·5-s + 2.54·6-s − 7-s − 6.30·8-s + 9-s − 1.66·10-s + 1.19·11-s − 4.47·12-s − 6.03·13-s + 2.54·14-s − 0.653·15-s + 7.08·16-s − 2.30·17-s − 2.54·18-s + 5.19·19-s + 2.92·20-s + 21-s − 3.03·22-s − 1.76·23-s + 6.30·24-s − 4.57·25-s + 15.3·26-s − 27-s − 4.47·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 0.577·3-s + 2.23·4-s + 0.292·5-s + 1.03·6-s − 0.377·7-s − 2.22·8-s + 0.333·9-s − 0.525·10-s + 0.359·11-s − 1.29·12-s − 1.67·13-s + 0.680·14-s − 0.168·15-s + 1.77·16-s − 0.557·17-s − 0.599·18-s + 1.19·19-s + 0.653·20-s + 0.218·21-s − 0.647·22-s − 0.368·23-s + 1.28·24-s − 0.914·25-s + 3.01·26-s − 0.192·27-s − 0.846·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 5 | \( 1 - 0.653T + 5T^{2} \) |
| 11 | \( 1 - 1.19T + 11T^{2} \) |
| 13 | \( 1 + 6.03T + 13T^{2} \) |
| 17 | \( 1 + 2.30T + 17T^{2} \) |
| 19 | \( 1 - 5.19T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 - 0.998T + 29T^{2} \) |
| 31 | \( 1 + 0.532T + 31T^{2} \) |
| 37 | \( 1 - 3.77T + 37T^{2} \) |
| 41 | \( 1 + 7.36T + 41T^{2} \) |
| 43 | \( 1 + 1.47T + 43T^{2} \) |
| 47 | \( 1 - 1.89T + 47T^{2} \) |
| 53 | \( 1 - 9.51T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 + 5.22T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 - 7.15T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 1.39T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 + 3.96T + 97T^{2} \) |
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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
−7.34906457564031541686592203805, −7.15091115537297907112150598604, −6.38827759686221431123531664331, −5.64866957041011760741066244742, −4.87800533391316055846040310636, −3.74165891539141443412567920688, −2.58780252228143324390960292783, −2.01185565894871015962445355054, −0.920157824400883514495339256152, 0,
0.920157824400883514495339256152, 2.01185565894871015962445355054, 2.58780252228143324390960292783, 3.74165891539141443412567920688, 4.87800533391316055846040310636, 5.64866957041011760741066244742, 6.38827759686221431123531664331, 7.15091115537297907112150598604, 7.34906457564031541686592203805
