L(s) = 1 | + 0.656·2-s + 2·3-s − 1.56·4-s + 3.25·5-s + 1.31·6-s − 2·7-s − 2.34·8-s + 9-s + 2.13·10-s − 5.48·11-s − 3.13·12-s + 5.13·13-s − 1.31·14-s + 6.51·15-s + 1.59·16-s + 2.34·17-s + 0.656·18-s − 5.73·19-s − 5.10·20-s − 4·21-s − 3.59·22-s + 1.31·23-s − 4.68·24-s + 5.59·25-s + 3.37·26-s − 4·27-s + 3.13·28-s + ⋯ |
L(s) = 1 | + 0.464·2-s + 1.15·3-s − 0.784·4-s + 1.45·5-s + 0.536·6-s − 0.755·7-s − 0.828·8-s + 0.333·9-s + 0.675·10-s − 1.65·11-s − 0.905·12-s + 1.42·13-s − 0.350·14-s + 1.68·15-s + 0.399·16-s + 0.568·17-s + 0.154·18-s − 1.31·19-s − 1.14·20-s − 0.872·21-s − 0.767·22-s + 0.273·23-s − 0.956·24-s + 1.11·25-s + 0.661·26-s − 0.769·27-s + 0.592·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.718708721\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.718708721\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 151 | \( 1 - T \) |
good | 2 | \( 1 - 0.656T + 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 - 3.25T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 5.48T + 11T^{2} \) |
| 13 | \( 1 - 5.13T + 13T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 19 | \( 1 + 5.73T + 19T^{2} \) |
| 23 | \( 1 - 1.31T + 23T^{2} \) |
| 29 | \( 1 + 4.42T + 29T^{2} \) |
| 31 | \( 1 - 2.56T + 31T^{2} \) |
| 37 | \( 1 + 0.969T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 + 2.08T + 47T^{2} \) |
| 53 | \( 1 - 11.4T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 14.2T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 + 1.31T + 71T^{2} \) |
| 73 | \( 1 - 9.13T + 73T^{2} \) |
| 79 | \( 1 + 2.86T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−13.33895001351975273717892205796, −12.75153133248952875070276237165, −10.64109079219684365159506337741, −9.763216389754272012425876758491, −8.926364582069228686377422534356, −8.119085420415903435014939465545, −6.22307463988278107453548334917, −5.36270242566125028937965701884, −3.61192629299886806621836297533, −2.46494606590211108372264941213,
2.46494606590211108372264941213, 3.61192629299886806621836297533, 5.36270242566125028937965701884, 6.22307463988278107453548334917, 8.119085420415903435014939465545, 8.926364582069228686377422534356, 9.763216389754272012425876758491, 10.64109079219684365159506337741, 12.75153133248952875070276237165, 13.33895001351975273717892205796