| L(s) = 1 | + 1.47·2-s − 0.565·3-s + 0.164·4-s + 3.06·5-s − 0.831·6-s − 0.0154·7-s − 2.70·8-s − 2.68·9-s + 4.51·10-s − 0.329·11-s − 0.0926·12-s + 2.08·13-s − 0.0226·14-s − 1.73·15-s − 4.30·16-s − 2.76·17-s − 3.94·18-s − 1.81·19-s + 0.503·20-s + 0.00871·21-s − 0.485·22-s − 6.26·23-s + 1.52·24-s + 4.41·25-s + 3.06·26-s + 3.21·27-s − 0.00252·28-s + ⋯ |
| L(s) = 1 | + 1.04·2-s − 0.326·3-s + 0.0820·4-s + 1.37·5-s − 0.339·6-s − 0.00582·7-s − 0.954·8-s − 0.893·9-s + 1.42·10-s − 0.0994·11-s − 0.0267·12-s + 0.577·13-s − 0.00606·14-s − 0.447·15-s − 1.07·16-s − 0.670·17-s − 0.929·18-s − 0.416·19-s + 0.112·20-s + 0.00190·21-s − 0.103·22-s − 1.30·23-s + 0.311·24-s + 0.882·25-s + 0.601·26-s + 0.617·27-s − 0.000477·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.494919036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.494919036\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.47T + 2T^{2} \) |
| 3 | \( 1 + 0.565T + 3T^{2} \) |
| 5 | \( 1 - 3.06T + 5T^{2} \) |
| 7 | \( 1 + 0.0154T + 7T^{2} \) |
| 11 | \( 1 + 0.329T + 11T^{2} \) |
| 13 | \( 1 - 2.08T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 + 6.26T + 23T^{2} \) |
| 29 | \( 1 - 8.49T + 29T^{2} \) |
| 31 | \( 1 + 0.492T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 + 4.67T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 - 1.88T + 53T^{2} \) |
| 59 | \( 1 + 9.22T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 + 6.34T + 67T^{2} \) |
| 71 | \( 1 - 7.05T + 71T^{2} \) |
| 73 | \( 1 + 5.97T + 73T^{2} \) |
| 79 | \( 1 + 4.80T + 79T^{2} \) |
| 83 | \( 1 + 2.21T + 83T^{2} \) |
| 89 | \( 1 + 8.80T + 89T^{2} \) |
| 97 | \( 1 + 6.36T + 97T^{2} \) |
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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
−13.84896052929753319806963393944, −13.02531490167780929462851464884, −11.97982768921089290758363730930, −10.79692523185914596051136816721, −9.548884071250594107126140866624, −8.497590476285771691151008605261, −6.24616544767485028564825366414, −5.84451317547634848546149775809, −4.48034791252540293684516339516, −2.63607602675867454928031563943,
2.63607602675867454928031563943, 4.48034791252540293684516339516, 5.84451317547634848546149775809, 6.24616544767485028564825366414, 8.497590476285771691151008605261, 9.548884071250594107126140866624, 10.79692523185914596051136816721, 11.97982768921089290758363730930, 13.02531490167780929462851464884, 13.84896052929753319806963393944
