| L(s) = 1 | + 0.339·3-s + 4.05·7-s − 2.88·9-s − 11-s + 4·13-s + 7.74·17-s + 7.06·19-s + 1.37·21-s + 2.72·23-s − 1.99·27-s − 4.73·29-s − 0.219·31-s − 0.339·33-s + 1.32·37-s + 1.35·39-s − 7.79·41-s − 11.1·43-s + 3.01·47-s + 9.42·49-s + 2.62·51-s + 5.03·53-s + 2.39·57-s − 10.9·59-s + 12.7·61-s − 11.6·63-s − 4.70·67-s + 0.924·69-s + ⋯ |
| L(s) = 1 | + 0.195·3-s + 1.53·7-s − 0.961·9-s − 0.301·11-s + 1.10·13-s + 1.87·17-s + 1.62·19-s + 0.299·21-s + 0.568·23-s − 0.384·27-s − 0.878·29-s − 0.0394·31-s − 0.0590·33-s + 0.218·37-s + 0.217·39-s − 1.21·41-s − 1.69·43-s + 0.439·47-s + 1.34·49-s + 0.367·51-s + 0.692·53-s + 0.317·57-s − 1.42·59-s + 1.62·61-s − 1.47·63-s − 0.574·67-s + 0.111·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.737429113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.737429113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 0.339T + 3T^{2} \) |
| 7 | \( 1 - 4.05T + 7T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 7.74T + 17T^{2} \) |
| 19 | \( 1 - 7.06T + 19T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 + 0.219T + 31T^{2} \) |
| 37 | \( 1 - 1.32T + 37T^{2} \) |
| 41 | \( 1 + 7.79T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 3.01T + 47T^{2} \) |
| 53 | \( 1 - 5.03T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 - 2.52T + 71T^{2} \) |
| 73 | \( 1 + 4.10T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 3.63T + 83T^{2} \) |
| 89 | \( 1 - 9.88T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.318306936707124662359387597153, −7.76671585777777043687094675186, −7.16533177521721817568933473481, −5.85153147642684920192136597939, −5.44612377046570981871933591819, −4.83451801633646439896245221857, −3.55661734458724636837531496530, −3.10690126702780991213280084628, −1.78823786800192115465228818986, −1.00744498420148429851199282256,
1.00744498420148429851199282256, 1.78823786800192115465228818986, 3.10690126702780991213280084628, 3.55661734458724636837531496530, 4.83451801633646439896245221857, 5.44612377046570981871933591819, 5.85153147642684920192136597939, 7.16533177521721817568933473481, 7.76671585777777043687094675186, 8.318306936707124662359387597153
