L(s) = 1 | − 2.64·2-s − 1.80·3-s + 5.00·4-s + 5-s + 4.78·6-s − 4.64·7-s − 7.94·8-s + 0.272·9-s − 2.64·10-s − 11-s − 9.04·12-s − 2.93·13-s + 12.2·14-s − 1.80·15-s + 11.0·16-s − 0.403·17-s − 0.720·18-s + 1.76·19-s + 5.00·20-s + 8.39·21-s + 2.64·22-s + 1.86·23-s + 14.3·24-s + 25-s + 7.77·26-s + 4.93·27-s − 23.2·28-s + ⋯ |
L(s) = 1 | − 1.87·2-s − 1.04·3-s + 2.50·4-s + 0.447·5-s + 1.95·6-s − 1.75·7-s − 2.80·8-s + 0.0907·9-s − 0.836·10-s − 0.301·11-s − 2.61·12-s − 0.815·13-s + 3.28·14-s − 0.467·15-s + 2.75·16-s − 0.0979·17-s − 0.169·18-s + 0.405·19-s + 1.11·20-s + 1.83·21-s + 0.564·22-s + 0.388·23-s + 2.93·24-s + 0.200·25-s + 1.52·26-s + 0.949·27-s − 4.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.03293324546\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03293324546\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 + 1.80T + 3T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 13 | \( 1 + 2.93T + 13T^{2} \) |
| 17 | \( 1 + 0.403T + 17T^{2} \) |
| 19 | \( 1 - 1.76T + 19T^{2} \) |
| 23 | \( 1 - 1.86T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 + 1.81T + 31T^{2} \) |
| 37 | \( 1 + 2.91T + 37T^{2} \) |
| 41 | \( 1 + 3.53T + 41T^{2} \) |
| 43 | \( 1 + 8.27T + 43T^{2} \) |
| 47 | \( 1 + 7.73T + 47T^{2} \) |
| 53 | \( 1 - 0.854T + 53T^{2} \) |
| 59 | \( 1 + 7.27T + 59T^{2} \) |
| 61 | \( 1 + 6.85T + 61T^{2} \) |
| 67 | \( 1 + 7.50T + 67T^{2} \) |
| 71 | \( 1 - 1.08T + 71T^{2} \) |
| 79 | \( 1 + 3.60T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 + 9.10T + 97T^{2} \) |
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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
−8.726893872764799277094993425226, −7.61661354567882488701760687858, −7.03767041993639650874145523064, −6.44821706142278847078477869944, −5.87219692240992944403807661131, −5.11319443099691832668766745551, −3.38098365789242096375389259730, −2.68470860622188577532915080653, −1.55901059038308855174831996344, −0.14181693807183959921971627300,
0.14181693807183959921971627300, 1.55901059038308855174831996344, 2.68470860622188577532915080653, 3.38098365789242096375389259730, 5.11319443099691832668766745551, 5.87219692240992944403807661131, 6.44821706142278847078477869944, 7.03767041993639650874145523064, 7.61661354567882488701760687858, 8.726893872764799277094993425226