L(s) = 1 | − 2.11·2-s − 5.68·3-s − 3.54·4-s − 12.6·5-s + 12.0·6-s + 24.3·8-s + 5.36·9-s + 26.6·10-s − 10.0·11-s + 20.1·12-s + 12.8·13-s + 71.9·15-s − 23.0·16-s − 50.2·17-s − 11.3·18-s − 42.4·19-s + 44.8·20-s + 21.1·22-s + 50.9·23-s − 138.·24-s + 34.8·25-s − 27.0·26-s + 123.·27-s − 177.·29-s − 151.·30-s + 261.·31-s − 146.·32-s + ⋯ |
L(s) = 1 | − 0.746·2-s − 1.09·3-s − 0.443·4-s − 1.13·5-s + 0.816·6-s + 1.07·8-s + 0.198·9-s + 0.843·10-s − 0.274·11-s + 0.485·12-s + 0.273·13-s + 1.23·15-s − 0.360·16-s − 0.716·17-s − 0.148·18-s − 0.512·19-s + 0.501·20-s + 0.204·22-s + 0.461·23-s − 1.17·24-s + 0.278·25-s − 0.204·26-s + 0.877·27-s − 1.13·29-s − 0.923·30-s + 1.51·31-s − 0.808·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1978953598\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1978953598\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + 2.11T + 8T^{2} \) |
| 3 | \( 1 + 5.68T + 27T^{2} \) |
| 5 | \( 1 + 12.6T + 125T^{2} \) |
| 11 | \( 1 + 10.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 12.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 50.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 42.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 50.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 177.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 261.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 88.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 408.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 479.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 424.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 336.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 642.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 500.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 796.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 296.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 508.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 897.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 185.T + 9.12e5T^{2} \) |
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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
−8.801138466178493568551675678545, −7.72806633260841525395213924893, −7.39304791634573352180074931266, −6.26584391163093602873808318977, −5.53921733509080757356456081092, −4.42424496822325294042362523037, −4.19888583149550331205110528252, −2.75154596703294437329390058516, −1.20060234780879595483360762920, −0.27519535005464903705556509262,
0.27519535005464903705556509262, 1.20060234780879595483360762920, 2.75154596703294437329390058516, 4.19888583149550331205110528252, 4.42424496822325294042362523037, 5.53921733509080757356456081092, 6.26584391163093602873808318977, 7.39304791634573352180074931266, 7.72806633260841525395213924893, 8.801138466178493568551675678545