| L(s) = 1 | + 2.86·3-s − 3.00·5-s + 3.89·7-s + 5.20·9-s + 2.75·11-s + 7.07·13-s − 8.60·15-s + 17-s + 3.96·19-s + 11.1·21-s + 2.50·23-s + 4.01·25-s + 6.32·27-s − 6.77·29-s + 10.7·31-s + 7.88·33-s − 11.7·35-s + 9.11·37-s + 20.2·39-s − 5.93·41-s − 2.16·43-s − 15.6·45-s + 2.01·47-s + 8.18·49-s + 2.86·51-s − 12.1·53-s − 8.25·55-s + ⋯ |
| L(s) = 1 | + 1.65·3-s − 1.34·5-s + 1.47·7-s + 1.73·9-s + 0.829·11-s + 1.96·13-s − 2.22·15-s + 0.242·17-s + 0.908·19-s + 2.43·21-s + 0.521·23-s + 0.802·25-s + 1.21·27-s − 1.25·29-s + 1.92·31-s + 1.37·33-s − 1.97·35-s + 1.49·37-s + 3.24·39-s − 0.926·41-s − 0.329·43-s − 2.33·45-s + 0.294·47-s + 1.16·49-s + 0.401·51-s − 1.66·53-s − 1.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.752155092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.752155092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 2.86T + 3T^{2} \) |
| 5 | \( 1 + 3.00T + 5T^{2} \) |
| 7 | \( 1 - 3.89T + 7T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 13 | \( 1 - 7.07T + 13T^{2} \) |
| 19 | \( 1 - 3.96T + 19T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 + 6.77T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 9.11T + 37T^{2} \) |
| 41 | \( 1 + 5.93T + 41T^{2} \) |
| 43 | \( 1 + 2.16T + 43T^{2} \) |
| 47 | \( 1 - 2.01T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 9.34T + 67T^{2} \) |
| 71 | \( 1 + 7.15T + 71T^{2} \) |
| 73 | \( 1 + 1.43T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 2.91T + 83T^{2} \) |
| 89 | \( 1 + 4.82T + 89T^{2} \) |
| 97 | \( 1 - 7.50T + 97T^{2} \) |
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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
−7.86782482581802170721011083546, −7.64853772193620679801254271703, −6.70943371668620726328751419892, −5.76458609719936924327310160320, −4.47625576157014612572382490378, −4.28378119512585485821328392617, −3.40418439803457153692004791685, −3.00052462103540587318102290897, −1.58459886728399685599926503921, −1.19855173953519108556141884330,
1.19855173953519108556141884330, 1.58459886728399685599926503921, 3.00052462103540587318102290897, 3.40418439803457153692004791685, 4.28378119512585485821328392617, 4.47625576157014612572382490378, 5.76458609719936924327310160320, 6.70943371668620726328751419892, 7.64853772193620679801254271703, 7.86782482581802170721011083546
