L(s) = 1 | − 2.70·2-s + 0.539·3-s + 5.34·4-s + 1.70·5-s − 1.46·6-s + 1.46·7-s − 9.04·8-s − 2.70·9-s − 4.63·10-s + 3.80·11-s + 2.87·12-s − 4.34·13-s − 3.95·14-s + 0.921·15-s + 13.8·16-s − 2·17-s + 7.34·18-s − 1.80·19-s + 9.12·20-s + 0.787·21-s − 10.2·22-s − 1.26·23-s − 4.87·24-s − 2.07·25-s + 11.7·26-s − 3.07·27-s + 7.80·28-s + ⋯ |
L(s) = 1 | − 1.91·2-s + 0.311·3-s + 2.67·4-s + 0.764·5-s − 0.596·6-s + 0.552·7-s − 3.19·8-s − 0.903·9-s − 1.46·10-s + 1.14·11-s + 0.831·12-s − 1.20·13-s − 1.05·14-s + 0.237·15-s + 3.45·16-s − 0.485·17-s + 1.73·18-s − 0.413·19-s + 2.04·20-s + 0.171·21-s − 2.19·22-s − 0.263·23-s − 0.995·24-s − 0.415·25-s + 2.30·26-s − 0.592·27-s + 1.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4437913410\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4437913410\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 - T \) |
good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 - 0.539T + 3T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 + 3.07T + 29T^{2} \) |
| 31 | \( 1 - 0.581T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 43 | \( 1 - 2.34T + 43T^{2} \) |
| 47 | \( 1 - 12.6T + 47T^{2} \) |
| 53 | \( 1 - 0.921T + 53T^{2} \) |
| 59 | \( 1 + 5.26T + 59T^{2} \) |
| 61 | \( 1 + 7.75T + 61T^{2} \) |
| 67 | \( 1 + 3.80T + 67T^{2} \) |
| 71 | \( 1 + 1.21T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 14.8T + 79T^{2} \) |
| 83 | \( 1 + 2.15T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 4.15T + 97T^{2} \) |
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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
−16.91026960860727447323384313641, −15.18945696241584830631417044128, −14.22253080403673396082387708383, −12.02697670630786306497922221880, −10.99349616235852862205907573115, −9.644025101467508164016452203628, −8.903833403877222114675285043380, −7.63196582345436066291380439632, −6.15594604372841359728162390431, −2.20228565748956291852367527713,
2.20228565748956291852367527713, 6.15594604372841359728162390431, 7.63196582345436066291380439632, 8.903833403877222114675285043380, 9.644025101467508164016452203628, 10.99349616235852862205907573115, 12.02697670630786306497922221880, 14.22253080403673396082387708383, 15.18945696241584830631417044128, 16.91026960860727447323384313641