| L(s) = 1 | − 1.62·2-s − 2.83·3-s − 5.37·4-s − 8.40·5-s + 4.60·6-s − 9.04·7-s + 21.6·8-s − 18.9·9-s + 13.6·10-s − 11·11-s + 15.2·12-s − 13·13-s + 14.6·14-s + 23.8·15-s + 7.80·16-s + 121.·17-s + 30.7·18-s + 74.4·19-s + 45.1·20-s + 25.6·21-s + 17.8·22-s − 159.·23-s − 61.5·24-s − 54.2·25-s + 21.0·26-s + 130.·27-s + 48.6·28-s + ⋯ |
| L(s) = 1 | − 0.573·2-s − 0.546·3-s − 0.671·4-s − 0.752·5-s + 0.313·6-s − 0.488·7-s + 0.958·8-s − 0.701·9-s + 0.431·10-s − 0.301·11-s + 0.366·12-s − 0.277·13-s + 0.280·14-s + 0.411·15-s + 0.121·16-s + 1.74·17-s + 0.402·18-s + 0.899·19-s + 0.504·20-s + 0.266·21-s + 0.172·22-s − 1.44·23-s − 0.523·24-s − 0.434·25-s + 0.159·26-s + 0.929·27-s + 0.328·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4961346679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4961346679\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 + 13T \) |
| good | 2 | \( 1 + 1.62T + 8T^{2} \) |
| 3 | \( 1 + 2.83T + 27T^{2} \) |
| 5 | \( 1 + 8.40T + 125T^{2} \) |
| 7 | \( 1 + 9.04T + 343T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 74.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 159.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 30.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 171.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 285.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 67.0T + 1.03e5T^{2} \) |
| 53 | \( 1 + 422.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 427.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 128.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 622.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 839.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 650.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 891.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 730.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.17e3T + 9.12e5T^{2} \) |
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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
−12.29871848725144000293523016003, −11.78091622465613353087295137490, −10.35821069563014992638452619847, −9.707000290150088647223230364025, −8.303729754516655159281476500844, −7.67063733558537119617634987194, −6.00954289401557594276498877635, −4.81866005936667896251053989780, −3.33220719879134967076334234762, −0.64344086283556782100807243566,
0.64344086283556782100807243566, 3.33220719879134967076334234762, 4.81866005936667896251053989780, 6.00954289401557594276498877635, 7.67063733558537119617634987194, 8.303729754516655159281476500844, 9.707000290150088647223230364025, 10.35821069563014992638452619847, 11.78091622465613353087295137490, 12.29871848725144000293523016003
