L(s) = 1 | + 1.41·5-s − 8·7-s + 11.3·11-s − 8i·13-s − 12.7i·17-s + 32i·19-s − 33.9i·23-s − 23·25-s − 43.8·29-s + 40·31-s − 11.3·35-s + 26i·37-s + 66.4i·41-s − 16i·43-s − 11.3i·47-s + ⋯ |
L(s) = 1 | + 0.282·5-s − 1.14·7-s + 1.02·11-s − 0.615i·13-s − 0.748i·17-s + 1.68i·19-s − 1.47i·23-s − 0.920·25-s − 1.51·29-s + 1.29·31-s − 0.323·35-s + 0.702i·37-s + 1.62i·41-s − 0.372i·43-s − 0.240i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03667752542\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03667752542\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 1.41T + 25T^{2} \) |
| 7 | \( 1 + 8T + 49T^{2} \) |
| 11 | \( 1 - 11.3T + 121T^{2} \) |
| 13 | \( 1 + 8iT - 169T^{2} \) |
| 17 | \( 1 + 12.7iT - 289T^{2} \) |
| 19 | \( 1 - 32iT - 361T^{2} \) |
| 23 | \( 1 + 33.9iT - 529T^{2} \) |
| 29 | \( 1 + 43.8T + 841T^{2} \) |
| 31 | \( 1 - 40T + 961T^{2} \) |
| 37 | \( 1 - 26iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 66.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 16iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 11.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 32.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 22.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 54iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 80iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 79.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 96T + 5.32e3T^{2} \) |
| 79 | \( 1 + 104T + 6.24e3T^{2} \) |
| 83 | \( 1 - 101.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 77.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 80T + 9.40e3T^{2} \) |
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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
−9.369773671871922505369854058771, −8.416759049353437500978055393875, −7.70341303724271353838874022206, −6.60458126636050941415000544718, −6.25373823218466162766484865047, −5.39816563331625518987754171549, −4.21997817161554947594306570506, −3.47850624896089051184017132241, −2.55977495382256667386514216578, −1.29518052050737481884793050177,
0.008980643404592978051554967899, 1.43393750777722667586831938109, 2.52443740819015770592049993572, 3.63843157890471106415011584868, 4.19511532519424435870255524045, 5.49153526887748362746881151699, 6.14647719220171940046115242553, 6.89873717083227407722932990951, 7.47208755448719434494998410274, 8.757419179723430416722830087848