| L(s) = 1 | − 2-s − 1.19·3-s + 4-s + 1.77·5-s + 1.19·6-s + 2.63·7-s − 8-s − 1.57·9-s − 1.77·10-s − 1.87·11-s − 1.19·12-s − 6.18·13-s − 2.63·14-s − 2.11·15-s + 16-s + 5.03·17-s + 1.57·18-s + 0.622·19-s + 1.77·20-s − 3.15·21-s + 1.87·22-s − 4.47·23-s + 1.19·24-s − 1.85·25-s + 6.18·26-s + 5.46·27-s + 2.63·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.689·3-s + 0.5·4-s + 0.793·5-s + 0.487·6-s + 0.997·7-s − 0.353·8-s − 0.524·9-s − 0.560·10-s − 0.564·11-s − 0.344·12-s − 1.71·13-s − 0.705·14-s − 0.546·15-s + 0.250·16-s + 1.22·17-s + 0.370·18-s + 0.142·19-s + 0.396·20-s − 0.687·21-s + 0.399·22-s − 0.933·23-s + 0.243·24-s − 0.370·25-s + 1.21·26-s + 1.05·27-s + 0.498·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 + 1.19T + 3T^{2} \) |
| 5 | \( 1 - 1.77T + 5T^{2} \) |
| 7 | \( 1 - 2.63T + 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 + 6.18T + 13T^{2} \) |
| 17 | \( 1 - 5.03T + 17T^{2} \) |
| 19 | \( 1 - 0.622T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 - 9.96T + 29T^{2} \) |
| 31 | \( 1 - 3.49T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 0.549T + 43T^{2} \) |
| 47 | \( 1 + 4.29T + 47T^{2} \) |
| 53 | \( 1 - 1.92T + 53T^{2} \) |
| 59 | \( 1 + 9.34T + 59T^{2} \) |
| 61 | \( 1 + 4.71T + 61T^{2} \) |
| 67 | \( 1 + 0.131T + 67T^{2} \) |
| 71 | \( 1 + 4.03T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 - 8.76T + 79T^{2} \) |
| 83 | \( 1 + 8.41T + 83T^{2} \) |
| 89 | \( 1 - 1.49T + 89T^{2} \) |
| 97 | \( 1 + 2.31T + 97T^{2} \) |
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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
−7.999631983998042413380125943206, −7.58592299885930499717434273932, −6.53770355721872738889350949420, −5.88252326300577218019383287657, −5.12417949230153008785645464674, −4.70480180723757293827279076976, −3.03513471837144107146986720542, −2.30611853496933796825859106676, −1.31104370257556885993441604442, 0,
1.31104370257556885993441604442, 2.30611853496933796825859106676, 3.03513471837144107146986720542, 4.70480180723757293827279076976, 5.12417949230153008785645464674, 5.88252326300577218019383287657, 6.53770355721872738889350949420, 7.58592299885930499717434273932, 7.999631983998042413380125943206
