L(s) = 1 | − 3.76·2-s − 4.70·3-s + 6.16·4-s + 20.6·5-s + 17.7·6-s − 28.6·7-s + 6.91·8-s − 4.84·9-s − 77.5·10-s − 11·11-s − 29.0·12-s − 13·13-s + 107.·14-s − 96.9·15-s − 75.3·16-s + 50.0·17-s + 18.2·18-s + 5.38·19-s + 126.·20-s + 134.·21-s + 41.3·22-s + 183.·23-s − 32.5·24-s + 299.·25-s + 48.9·26-s + 149.·27-s − 176.·28-s + ⋯ |
L(s) = 1 | − 1.33·2-s − 0.905·3-s + 0.770·4-s + 1.84·5-s + 1.20·6-s − 1.54·7-s + 0.305·8-s − 0.179·9-s − 2.45·10-s − 0.301·11-s − 0.697·12-s − 0.277·13-s + 2.05·14-s − 1.66·15-s − 1.17·16-s + 0.714·17-s + 0.238·18-s + 0.0649·19-s + 1.41·20-s + 1.40·21-s + 0.401·22-s + 1.66·23-s − 0.276·24-s + 2.39·25-s + 0.369·26-s + 1.06·27-s − 1.19·28-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
\(L(2)\) |
\(\approx\) |
\(0.6106952373\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6106952373\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.76T + 8T^{2} \) |
| 3 | \( 1 + 4.70T + 27T^{2} \) |
| 5 | \( 1 - 20.6T + 125T^{2} \) |
| 7 | \( 1 + 28.6T + 343T^{2} \) |
| 17 | \( 1 - 50.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.38T + 6.85e3T^{2} \) |
| 23 | \( 1 - 183.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 200.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 19.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 359.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 261.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 450.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 725.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 381.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 68.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 133.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 142.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 394.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.15e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 519.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 248.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 848.T + 9.12e5T^{2} \) |
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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
−12.79872557901009462149002972624, −11.14718439740545620448512178568, −10.26408040770734570476139762547, −9.605145464597633201921554802636, −8.987485766549773729195495962588, −7.16783499178004325396712809682, −6.18643580730669169585960002866, −5.34044784485604815028863313873, −2.63249420337107675747850241871, −0.817296351662831924339291380066,
0.817296351662831924339291380066, 2.63249420337107675747850241871, 5.34044784485604815028863313873, 6.18643580730669169585960002866, 7.16783499178004325396712809682, 8.987485766549773729195495962588, 9.605145464597633201921554802636, 10.26408040770734570476139762547, 11.14718439740545620448512178568, 12.79872557901009462149002972624