L(s) = 1 | − 0.789·2-s − 3·3-s − 7.37·4-s + 13.9·5-s + 2.36·6-s − 21.1·7-s + 12.1·8-s + 9·9-s − 10.9·10-s − 11·11-s + 22.1·12-s + 41.9·13-s + 16.6·14-s − 41.7·15-s + 49.4·16-s + 53.5·17-s − 7.10·18-s + 133.·19-s − 102.·20-s + 63.4·21-s + 8.68·22-s − 11.7·23-s − 36.4·24-s + 69.0·25-s − 33.1·26-s − 27·27-s + 156.·28-s + ⋯ |
L(s) = 1 | − 0.279·2-s − 0.577·3-s − 0.922·4-s + 1.24·5-s + 0.161·6-s − 1.14·7-s + 0.536·8-s + 0.333·9-s − 0.347·10-s − 0.301·11-s + 0.532·12-s + 0.895·13-s + 0.318·14-s − 0.719·15-s + 0.772·16-s + 0.763·17-s − 0.0930·18-s + 1.61·19-s − 1.14·20-s + 0.659·21-s + 0.0841·22-s − 0.106·23-s − 0.309·24-s + 0.552·25-s − 0.249·26-s − 0.192·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.418312628\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.418312628\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
good | 2 | \( 1 + 0.789T + 8T^{2} \) |
| 5 | \( 1 - 13.9T + 125T^{2} \) |
| 7 | \( 1 + 21.1T + 343T^{2} \) |
| 13 | \( 1 - 41.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 53.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 133.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 11.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 184.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 330.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 449.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 384.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 324.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 406.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 528.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 455.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 991.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 574.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 317.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.49e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 346.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 317.T + 9.12e5T^{2} \) |
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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
−9.107039368300916823706249622936, −8.126827336939912888510270666941, −7.14601988983073029867401085291, −6.24047085761404277461686341339, −5.61214900770053954254397189291, −5.06107498603109171277791519055, −3.79281946255061921592530060211, −2.98149239752710175370080975960, −1.48379818994082871827719105482, −0.64480620402963660123922626835,
0.64480620402963660123922626835, 1.48379818994082871827719105482, 2.98149239752710175370080975960, 3.79281946255061921592530060211, 5.06107498603109171277791519055, 5.61214900770053954254397189291, 6.24047085761404277461686341339, 7.14601988983073029867401085291, 8.126827336939912888510270666941, 9.107039368300916823706249622936