| L(s) = 1 | + 0.651·2-s − 0.506·3-s − 1.57·4-s − 2.90·5-s − 0.330·6-s − 3.71·7-s − 2.32·8-s − 2.74·9-s − 1.89·10-s − 11-s + 0.798·12-s + 6.09·13-s − 2.41·14-s + 1.47·15-s + 1.63·16-s − 4.76·17-s − 1.78·18-s − 5.31·19-s + 4.57·20-s + 1.88·21-s − 0.651·22-s + 3.91·23-s + 1.18·24-s + 3.43·25-s + 3.96·26-s + 2.90·27-s + 5.84·28-s + ⋯ |
| L(s) = 1 | + 0.460·2-s − 0.292·3-s − 0.787·4-s − 1.29·5-s − 0.134·6-s − 1.40·7-s − 0.823·8-s − 0.914·9-s − 0.598·10-s − 0.301·11-s + 0.230·12-s + 1.68·13-s − 0.646·14-s + 0.379·15-s + 0.408·16-s − 1.15·17-s − 0.421·18-s − 1.21·19-s + 1.02·20-s + 0.410·21-s − 0.138·22-s + 0.816·23-s + 0.240·24-s + 0.686·25-s + 0.778·26-s + 0.559·27-s + 1.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3773039593\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3773039593\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 - 0.651T + 2T^{2} \) |
| 3 | \( 1 + 0.506T + 3T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 + 3.71T + 7T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 + 4.76T + 17T^{2} \) |
| 19 | \( 1 + 5.31T + 19T^{2} \) |
| 23 | \( 1 - 3.91T + 23T^{2} \) |
| 29 | \( 1 + 0.423T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 0.881T + 37T^{2} \) |
| 41 | \( 1 + 6.61T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 7.72T + 47T^{2} \) |
| 53 | \( 1 - 6.48T + 53T^{2} \) |
| 59 | \( 1 - 3.06T + 59T^{2} \) |
| 61 | \( 1 + 8.41T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 0.442T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−9.084027522606744144633678095739, −8.921838024726164848240193874445, −8.096069174576694493992624775023, −6.89283843175649565706723229953, −6.15702994871517365395709833469, −5.42451639009072066599661742139, −4.13190354146041340762201408594, −3.75289552285462247225138192628, −2.83380007309512237852481078916, −0.39475758018661717483624090004,
0.39475758018661717483624090004, 2.83380007309512237852481078916, 3.75289552285462247225138192628, 4.13190354146041340762201408594, 5.42451639009072066599661742139, 6.15702994871517365395709833469, 6.89283843175649565706723229953, 8.096069174576694493992624775023, 8.921838024726164848240193874445, 9.084027522606744144633678095739
