L(s) = 1 | − 2.77·2-s + 1.86·3-s + 5.70·4-s + 1.50·5-s − 5.18·6-s − 3.04·7-s − 10.2·8-s + 0.487·9-s − 4.17·10-s − 4.97·11-s + 10.6·12-s + 4.84·13-s + 8.45·14-s + 2.81·15-s + 17.1·16-s − 1.65·17-s − 1.35·18-s + 7.46·19-s + 8.58·20-s − 5.69·21-s + 13.8·22-s − 5.06·23-s − 19.1·24-s − 2.73·25-s − 13.4·26-s − 4.69·27-s − 17.3·28-s + ⋯ |
L(s) = 1 | − 1.96·2-s + 1.07·3-s + 2.85·4-s + 0.673·5-s − 2.11·6-s − 1.15·7-s − 3.63·8-s + 0.162·9-s − 1.32·10-s − 1.49·11-s + 3.07·12-s + 1.34·13-s + 2.26·14-s + 0.726·15-s + 4.27·16-s − 0.401·17-s − 0.319·18-s + 1.71·19-s + 1.92·20-s − 1.24·21-s + 2.94·22-s − 1.05·23-s − 3.91·24-s − 0.546·25-s − 2.63·26-s − 0.902·27-s − 3.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
good | 2 | \( 1 + 2.77T + 2T^{2} \) |
| 3 | \( 1 - 1.86T + 3T^{2} \) |
| 5 | \( 1 - 1.50T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 + 4.97T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 - 7.46T + 19T^{2} \) |
| 23 | \( 1 + 5.06T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 - 7.56T + 31T^{2} \) |
| 41 | \( 1 - 9.30T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 + 3.71T + 47T^{2} \) |
| 53 | \( 1 - 2.18T + 53T^{2} \) |
| 59 | \( 1 + 7.82T + 59T^{2} \) |
| 61 | \( 1 - 6.93T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 9.15T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 6.00T + 79T^{2} \) |
| 83 | \( 1 - 17.2T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 + 7.08T + 97T^{2} \) |
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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
−8.014507051430213273027725030408, −7.87527331922184731510180910427, −6.95254294247894704518691594277, −5.98215185601152698262086270372, −5.74219052763830118307949936525, −3.64447206847212836950550111579, −2.85683246733942519083734318852, −2.43193691210452903199333870467, −1.34122368993314103928333799988, 0,
1.34122368993314103928333799988, 2.43193691210452903199333870467, 2.85683246733942519083734318852, 3.64447206847212836950550111579, 5.74219052763830118307949936525, 5.98215185601152698262086270372, 6.95254294247894704518691594277, 7.87527331922184731510180910427, 8.014507051430213273027725030408