L(s) = 1 | + 1.75·2-s + 2.25·3-s + 1.06·4-s − 5-s + 3.94·6-s − 1.30·7-s − 1.62·8-s + 2.07·9-s − 1.75·10-s + 11-s + 2.41·12-s − 5.89·13-s − 2.28·14-s − 2.25·15-s − 4.99·16-s − 5.97·17-s + 3.63·18-s + 7.88·19-s − 1.06·20-s − 2.93·21-s + 1.75·22-s + 0.168·23-s − 3.67·24-s + 25-s − 10.3·26-s − 2.07·27-s − 1.39·28-s + ⋯ |
L(s) = 1 | + 1.23·2-s + 1.30·3-s + 0.534·4-s − 0.447·5-s + 1.61·6-s − 0.491·7-s − 0.576·8-s + 0.692·9-s − 0.554·10-s + 0.301·11-s + 0.695·12-s − 1.63·13-s − 0.609·14-s − 0.581·15-s − 1.24·16-s − 1.45·17-s + 0.857·18-s + 1.80·19-s − 0.239·20-s − 0.640·21-s + 0.373·22-s + 0.0351·23-s − 0.749·24-s + 0.200·25-s − 2.02·26-s − 0.400·27-s − 0.263·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
good | 2 | \( 1 - 1.75T + 2T^{2} \) |
| 3 | \( 1 - 2.25T + 3T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 13 | \( 1 + 5.89T + 13T^{2} \) |
| 17 | \( 1 + 5.97T + 17T^{2} \) |
| 19 | \( 1 - 7.88T + 19T^{2} \) |
| 23 | \( 1 - 0.168T + 23T^{2} \) |
| 29 | \( 1 + 6.66T + 29T^{2} \) |
| 31 | \( 1 + 6.68T + 31T^{2} \) |
| 37 | \( 1 - 2.98T + 37T^{2} \) |
| 41 | \( 1 + 8.29T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 8.53T + 53T^{2} \) |
| 59 | \( 1 + 3.78T + 59T^{2} \) |
| 61 | \( 1 - 4.96T + 61T^{2} \) |
| 67 | \( 1 + 0.419T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 79 | \( 1 - 1.00T + 79T^{2} \) |
| 83 | \( 1 - 8.50T + 83T^{2} \) |
| 89 | \( 1 + 6.42T + 89T^{2} \) |
| 97 | \( 1 + 3.47T + 97T^{2} \) |
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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
−7.987195002587354826566758087045, −7.20228703617469433916875526340, −6.79035573104781738622205353473, −5.53499468663803186027822795737, −4.99056217947360720043303945048, −4.01517749993617847153332431740, −3.51817293982768271493641975023, −2.77239429039751671642250782047, −2.07477422908008832010168084120, 0,
2.07477422908008832010168084120, 2.77239429039751671642250782047, 3.51817293982768271493641975023, 4.01517749993617847153332431740, 4.99056217947360720043303945048, 5.53499468663803186027822795737, 6.79035573104781738622205353473, 7.20228703617469433916875526340, 7.987195002587354826566758087045