| L(s) = 1 | + 2·4-s − 5.19i·7-s + 4·16-s + 3.46i·19-s + 5·25-s − 10.3i·28-s − 8.66i·31-s − 6.92i·37-s − 13·43-s − 20·49-s + 13·61-s + 8·64-s − 12.1i·67-s − 1.73i·73-s + 6.92i·76-s + ⋯ |
| L(s) = 1 | + 4-s − 1.96i·7-s + 16-s + 0.794i·19-s + 25-s − 1.96i·28-s − 1.55i·31-s − 1.13i·37-s − 1.98·43-s − 2.85·49-s + 1.66·61-s + 64-s − 1.48i·67-s − 0.202i·73-s + 0.794i·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.101818990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.101818990\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 5.19iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 13T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 1.73iT - 73T^{2} \) |
| 79 | \( 1 - 13T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 19.0iT - 97T^{2} \) |
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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
−9.562659418411670609560770303852, −8.169273316086556433005792833824, −7.63591482532445221823106186314, −6.89467014712338490590698282284, −6.32330957485005699752075428604, −5.14498838884368396271322134815, −4.01797065380307853057820167715, −3.36476624665477288762519080122, −1.99078763810558088228352808591, −0.823052652157452310745906884638,
1.60843659544344408192544238218, 2.63157065887121620480903494219, 3.20749209732832894567563078283, 4.94242111528745808001406511268, 5.50415515788240422222198479319, 6.50618795599704138466893898084, 6.95134974112846520078597538902, 8.301313944502768042012743912665, 8.634963774147364067892326096121, 9.606834570620730067007222022531
