| L(s) = 1 | − 2.68·3-s − 5-s + 3.20·7-s + 4.20·9-s + 6.44·11-s + 3.76·13-s + 2.68·15-s + 5.65·17-s − 19-s − 8.60·21-s + 7.20·23-s + 25-s − 3.24·27-s − 2.12·29-s + 10.8·31-s − 17.3·33-s − 3.20·35-s − 1.89·37-s − 10.1·39-s − 6.48·41-s − 5.49·43-s − 4.20·45-s − 2.86·47-s + 3.28·49-s − 15.1·51-s − 3.60·53-s − 6.44·55-s + ⋯ |
| L(s) = 1 | − 1.54·3-s − 0.447·5-s + 1.21·7-s + 1.40·9-s + 1.94·11-s + 1.04·13-s + 0.693·15-s + 1.37·17-s − 0.229·19-s − 1.87·21-s + 1.50·23-s + 0.200·25-s − 0.623·27-s − 0.395·29-s + 1.95·31-s − 3.01·33-s − 0.542·35-s − 0.311·37-s − 1.61·39-s − 1.01·41-s − 0.837·43-s − 0.627·45-s − 0.417·47-s + 0.469·49-s − 2.12·51-s − 0.495·53-s − 0.869·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.585690136\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.585690136\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.68T + 3T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 - 6.44T + 11T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 23 | \( 1 - 7.20T + 23T^{2} \) |
| 29 | \( 1 + 2.12T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 1.89T + 37T^{2} \) |
| 41 | \( 1 + 6.48T + 41T^{2} \) |
| 43 | \( 1 + 5.49T + 43T^{2} \) |
| 47 | \( 1 + 2.86T + 47T^{2} \) |
| 53 | \( 1 + 3.60T + 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 6.44T + 61T^{2} \) |
| 67 | \( 1 - 6.62T + 67T^{2} \) |
| 71 | \( 1 - 4.41T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 9.78T + 79T^{2} \) |
| 83 | \( 1 + 1.49T + 83T^{2} \) |
| 89 | \( 1 + 3.94T + 89T^{2} \) |
| 97 | \( 1 + 5.09T + 97T^{2} \) |
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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
−8.562646275776447287454717080111, −8.039996912063082710180604872271, −6.81188350236747858156981031782, −6.59381348807720979146696823803, −5.59014830806570694585308208695, −4.94939376608863781529137762204, −4.21283183523785510363162492553, −3.37265983826205096697019155906, −1.40120941377115658260478596404, −1.02552313335756724967161731382,
1.02552313335756724967161731382, 1.40120941377115658260478596404, 3.37265983826205096697019155906, 4.21283183523785510363162492553, 4.94939376608863781529137762204, 5.59014830806570694585308208695, 6.59381348807720979146696823803, 6.81188350236747858156981031782, 8.039996912063082710180604872271, 8.562646275776447287454717080111
