| L(s) = 1 | − 2.83·2-s − 1.09·3-s + 0.0114·4-s + 1.75·5-s + 3.10·6-s + 24.1·7-s + 22.6·8-s − 25.7·9-s − 4.97·10-s − 54.3·11-s − 0.0125·12-s + 16.0·13-s − 68.2·14-s − 1.93·15-s − 64.0·16-s − 36.8·17-s + 73.0·18-s − 95.7·19-s + 0.0201·20-s − 26.4·21-s + 153.·22-s − 45.5·23-s − 24.8·24-s − 121.·25-s − 45.5·26-s + 57.9·27-s + 0.275·28-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.211·3-s + 0.00142·4-s + 0.157·5-s + 0.211·6-s + 1.30·7-s + 0.999·8-s − 0.955·9-s − 0.157·10-s − 1.48·11-s − 0.000302·12-s + 0.342·13-s − 1.30·14-s − 0.0332·15-s − 1.00·16-s − 0.525·17-s + 0.956·18-s − 1.15·19-s + 0.000224·20-s − 0.275·21-s + 1.49·22-s − 0.412·23-s − 0.211·24-s − 0.975·25-s − 0.343·26-s + 0.413·27-s + 0.00186·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 - 103T \) |
| good | 2 | \( 1 + 2.83T + 8T^{2} \) |
| 3 | \( 1 + 1.09T + 27T^{2} \) |
| 5 | \( 1 - 1.75T + 125T^{2} \) |
| 7 | \( 1 - 24.1T + 343T^{2} \) |
| 11 | \( 1 + 54.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 36.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 45.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 149.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 131.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 272.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 276.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 267.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 727.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 374.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 149.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 611.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 902.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 223.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 631.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 981.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.79e3T + 9.12e5T^{2} \) |
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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
−12.84742134031155193109369058683, −11.08862314767089102818022996990, −10.89104319272443490533172103481, −9.419843287345597105608185974200, −8.268230203629679714992218659644, −7.76900047919008116117186624088, −5.78338595602035500920610869016, −4.56275763627242418710316997255, −2.06738763030278985555188863687, 0,
2.06738763030278985555188863687, 4.56275763627242418710316997255, 5.78338595602035500920610869016, 7.76900047919008116117186624088, 8.268230203629679714992218659644, 9.419843287345597105608185974200, 10.89104319272443490533172103481, 11.08862314767089102818022996990, 12.84742134031155193109369058683
