| L(s) = 1 | + 7-s + 3·11-s − 13-s + 3·17-s + 7·19-s + 3·23-s − 5·25-s − 2·31-s + 37-s + 6·41-s + 4·43-s + 6·47-s − 6·49-s − 9·53-s − 10·61-s − 2·67-s + 12·71-s + 5·73-s + 3·77-s − 2·79-s + 3·83-s + 3·89-s − 91-s + 2·97-s − 6·101-s − 14·103-s + 9·107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.904·11-s − 0.277·13-s + 0.727·17-s + 1.60·19-s + 0.625·23-s − 25-s − 0.359·31-s + 0.164·37-s + 0.937·41-s + 0.609·43-s + 0.875·47-s − 6/7·49-s − 1.23·53-s − 1.28·61-s − 0.244·67-s + 1.42·71-s + 0.585·73-s + 0.341·77-s − 0.225·79-s + 0.329·83-s + 0.317·89-s − 0.104·91-s + 0.203·97-s − 0.597·101-s − 1.37·103-s + 0.870·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.370595210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.370595210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.933357460865082450152553287023, −7.63004775598607836026748684699, −6.81723552877082144009830496387, −5.95418375561756223920308930451, −5.33180780022695947151433280147, −4.52243214467070757504523538163, −3.66835760316659774614958628600, −2.93167487990336811723047506998, −1.75772521738150068774751035784, −0.883234769599793187450506494618,
0.883234769599793187450506494618, 1.75772521738150068774751035784, 2.93167487990336811723047506998, 3.66835760316659774614958628600, 4.52243214467070757504523538163, 5.33180780022695947151433280147, 5.95418375561756223920308930451, 6.81723552877082144009830496387, 7.63004775598607836026748684699, 7.933357460865082450152553287023
