| L(s) = 1 | − 3-s − 5-s + 1.71·7-s + 9-s − 2.52·11-s − 6.72·13-s + 15-s + 4.44·17-s + 3.10·19-s − 1.71·21-s − 23-s + 25-s − 27-s + 5.01·29-s + 1.13·31-s + 2.52·33-s − 1.71·35-s + 2.49·37-s + 6.72·39-s − 7.39·41-s + 6.78·43-s − 45-s + 13.3·47-s − 4.07·49-s − 4.44·51-s − 12.4·53-s + 2.52·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.646·7-s + 0.333·9-s − 0.761·11-s − 1.86·13-s + 0.258·15-s + 1.07·17-s + 0.711·19-s − 0.373·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s + 0.932·29-s + 0.203·31-s + 0.439·33-s − 0.289·35-s + 0.410·37-s + 1.07·39-s − 1.15·41-s + 1.03·43-s − 0.149·45-s + 1.94·47-s − 0.581·49-s − 0.621·51-s − 1.70·53-s + 0.340·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 + 2.52T + 11T^{2} \) |
| 13 | \( 1 + 6.72T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 - 3.10T + 19T^{2} \) |
| 29 | \( 1 - 5.01T + 29T^{2} \) |
| 31 | \( 1 - 1.13T + 31T^{2} \) |
| 37 | \( 1 - 2.49T + 37T^{2} \) |
| 41 | \( 1 + 7.39T + 41T^{2} \) |
| 43 | \( 1 - 6.78T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 + 12.4T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 2.89T + 61T^{2} \) |
| 67 | \( 1 - 0.0836T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 5.57T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 1.97T + 83T^{2} \) |
| 89 | \( 1 + 7.62T + 89T^{2} \) |
| 97 | \( 1 - 9.83T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.63430124942084123580263509458, −7.36532116527335794893382330537, −6.34933018486822462938110348275, −5.41283413927970937533299475125, −4.96586672763093458840562415882, −4.35662415081029902359188600206, −3.17768872231806716615204314090, −2.41242431322483931380709153845, −1.18653349009750679583664561685, 0,
1.18653349009750679583664561685, 2.41242431322483931380709153845, 3.17768872231806716615204314090, 4.35662415081029902359188600206, 4.96586672763093458840562415882, 5.41283413927970937533299475125, 6.34933018486822462938110348275, 7.36532116527335794893382330537, 7.63430124942084123580263509458
