L(s) = 1 | + 4.28·2-s + 8.91·3-s + 10.3·4-s + 18.9·5-s + 38.2·6-s + 10.2·8-s + 52.4·9-s + 81.3·10-s + 2.99·11-s + 92.5·12-s − 36.2·13-s + 169.·15-s − 39.2·16-s + 57.8·17-s + 225.·18-s + 62.8·19-s + 197.·20-s + 12.8·22-s + 15.3·23-s + 91.0·24-s + 235.·25-s − 155.·26-s + 227.·27-s + 44.3·29-s + 725.·30-s − 272.·31-s − 250.·32-s + ⋯ |
L(s) = 1 | + 1.51·2-s + 1.71·3-s + 1.29·4-s + 1.69·5-s + 2.60·6-s + 0.451·8-s + 1.94·9-s + 2.57·10-s + 0.0819·11-s + 2.22·12-s − 0.772·13-s + 2.91·15-s − 0.613·16-s + 0.825·17-s + 2.94·18-s + 0.758·19-s + 2.20·20-s + 0.124·22-s + 0.139·23-s + 0.774·24-s + 1.88·25-s − 1.17·26-s + 1.61·27-s + 0.283·29-s + 4.41·30-s − 1.57·31-s − 1.38·32-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
\(L(2)\) |
\(\approx\) |
\(14.76268901\) |
\(L(\frac12)\) |
\(\approx\) |
\(14.76268901\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.28T + 8T^{2} \) |
| 3 | \( 1 - 8.91T + 27T^{2} \) |
| 5 | \( 1 - 18.9T + 125T^{2} \) |
| 11 | \( 1 - 2.99T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 57.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 62.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 15.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 44.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 272.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 111.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 53.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 170.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 464.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 802.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 71.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 223.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 153.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 935.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 279.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 273.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.46e3T + 9.12e5T^{2} \) |
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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
−8.848421212942872898706537280930, −7.61553627520695840846769526868, −7.04425534080104835093850027498, −6.03215712108143933416025883605, −5.37704261795392681559270541419, −4.60568484171082495145609845528, −3.54187858700004797523575486322, −2.91286035549626132522643546063, −2.24732577372372223117587357876, −1.48513045488692119166644592980,
1.48513045488692119166644592980, 2.24732577372372223117587357876, 2.91286035549626132522643546063, 3.54187858700004797523575486322, 4.60568484171082495145609845528, 5.37704261795392681559270541419, 6.03215712108143933416025883605, 7.04425534080104835093850027498, 7.61553627520695840846769526868, 8.848421212942872898706537280930