L(s) = 1 | − 5.39·2-s + 21.1·4-s − 13.0·5-s − 6.42·7-s − 70.7·8-s + 70.3·10-s − 26.2·11-s + 34.6·14-s + 212.·16-s + 123.·17-s + 109.·19-s − 275.·20-s + 141.·22-s + 63.4·23-s + 45.0·25-s − 135.·28-s − 225.·29-s + 200.·31-s − 582.·32-s − 668.·34-s + 83.7·35-s + 252.·37-s − 591.·38-s + 922.·40-s − 227.·41-s − 384.·43-s − 553.·44-s + ⋯ |
L(s) = 1 | − 1.90·2-s + 2.63·4-s − 1.16·5-s − 0.346·7-s − 3.12·8-s + 2.22·10-s − 0.718·11-s + 0.661·14-s + 3.32·16-s + 1.76·17-s + 1.32·19-s − 3.07·20-s + 1.37·22-s + 0.574·23-s + 0.360·25-s − 0.915·28-s − 1.44·29-s + 1.16·31-s − 3.21·32-s − 3.37·34-s + 0.404·35-s + 1.12·37-s − 2.52·38-s + 3.64·40-s − 0.866·41-s − 1.36·43-s − 1.89·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4589548866\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4589548866\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 5.39T + 8T^{2} \) |
| 5 | \( 1 + 13.0T + 125T^{2} \) |
| 7 | \( 1 + 6.42T + 343T^{2} \) |
| 11 | \( 1 + 26.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 123.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 109.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 63.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 225.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 200.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 252.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 227.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 384.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 34.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 61.0T + 1.48e5T^{2} \) |
| 59 | \( 1 - 80.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 26.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 931.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 427.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 108.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 384.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 85.9T + 5.71e5T^{2} \) |
| 89 | \( 1 - 495.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 190.T + 9.12e5T^{2} \) |
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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
−9.111619103897954872948884029448, −8.143581210558227038957068773226, −7.71216867911575858162175602716, −7.26403498506267576678133556283, −6.18854274234717449395729950452, −5.17582583339305853978405153033, −3.47275882137356083105717394382, −2.90718879027924682294491916107, −1.41712725266854264927426848139, −0.46950192917112332773387893470,
0.46950192917112332773387893470, 1.41712725266854264927426848139, 2.90718879027924682294491916107, 3.47275882137356083105717394382, 5.17582583339305853978405153033, 6.18854274234717449395729950452, 7.26403498506267576678133556283, 7.71216867911575858162175602716, 8.143581210558227038957068773226, 9.111619103897954872948884029448