| L(s) = 1 | + 3.38·3-s − 5-s − 3.56·7-s + 8.48·9-s − 0.809·11-s − 2.35·13-s − 3.38·15-s − 0.960·17-s − 1.53·19-s − 12.0·21-s + 1.98·23-s + 25-s + 18.5·27-s − 7.90·29-s − 1.33·31-s − 2.74·33-s + 3.56·35-s + 4.01·37-s − 7.96·39-s − 1.24·41-s − 8.20·43-s − 8.48·45-s − 4.10·47-s + 5.71·49-s − 3.25·51-s − 3.52·53-s + 0.809·55-s + ⋯ |
| L(s) = 1 | + 1.95·3-s − 0.447·5-s − 1.34·7-s + 2.82·9-s − 0.243·11-s − 0.652·13-s − 0.875·15-s − 0.232·17-s − 0.351·19-s − 2.63·21-s + 0.413·23-s + 0.200·25-s + 3.57·27-s − 1.46·29-s − 0.239·31-s − 0.477·33-s + 0.602·35-s + 0.660·37-s − 1.27·39-s − 0.194·41-s − 1.25·43-s − 1.26·45-s − 0.598·47-s + 0.816·49-s − 0.455·51-s − 0.484·53-s + 0.109·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 3.38T + 3T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 0.809T + 11T^{2} \) |
| 13 | \( 1 + 2.35T + 13T^{2} \) |
| 17 | \( 1 + 0.960T + 17T^{2} \) |
| 19 | \( 1 + 1.53T + 19T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 + 7.90T + 29T^{2} \) |
| 31 | \( 1 + 1.33T + 31T^{2} \) |
| 37 | \( 1 - 4.01T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 8.20T + 43T^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 + 3.52T + 53T^{2} \) |
| 59 | \( 1 + 0.147T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 - 2.13T + 71T^{2} \) |
| 73 | \( 1 - 1.71T + 73T^{2} \) |
| 79 | \( 1 + 9.57T + 79T^{2} \) |
| 83 | \( 1 + 5.07T + 83T^{2} \) |
| 89 | \( 1 - 10.5T + 89T^{2} \) |
| 97 | \( 1 - 4.21T + 97T^{2} \) |
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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
−7.67305418885364256863874531854, −6.93581289940237125773004717208, −6.49813276393336604186705814777, −5.23296731904017608392634646899, −4.30528412394355796523547096170, −3.68702710366768693205654728808, −3.07394329390184130711697057940, −2.51946506479633367962711381341, −1.58257707798888445493154716534, 0,
1.58257707798888445493154716534, 2.51946506479633367962711381341, 3.07394329390184130711697057940, 3.68702710366768693205654728808, 4.30528412394355796523547096170, 5.23296731904017608392634646899, 6.49813276393336604186705814777, 6.93581289940237125773004717208, 7.67305418885364256863874531854
