| L(s) = 1 | + 3.16i·3-s − 2.23·5-s − 4.24i·7-s − 7.00·9-s − 7.07i·15-s + 13.4·21-s − 1.41i·23-s + 5.00·25-s − 12.6i·27-s + 8.94·29-s + 9.48i·35-s + 12·41-s − 3.16i·43-s + 15.6·45-s − 9.89i·47-s + ⋯ |
| L(s) = 1 | + 1.82i·3-s − 0.999·5-s − 1.60i·7-s − 2.33·9-s − 1.82i·15-s + 2.92·21-s − 0.294i·23-s + 1.00·25-s − 2.43i·27-s + 1.66·29-s + 1.60i·35-s + 1.87·41-s − 0.482i·43-s + 2.33·45-s − 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.027275022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.027275022\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| good | 3 | \( 1 - 3.16iT - 3T^{2} \) |
| 7 | \( 1 + 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 - 8.94T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 3.16iT - 43T^{2} \) |
| 47 | \( 1 + 9.89iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 9.48iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.855144205437622563538225503055, −8.961776882060084361157049560092, −8.166613311007317049402579578116, −7.36314815985794068151360536411, −6.31788596532155351043484214172, −4.97326978725078764107098888676, −4.38500888007087595172946702715, −3.81148480680400456797114359294, −3.00573131074610767724235944884, −0.52184243415511880416879260795,
1.08467039362434501894254952975, 2.41223712464075483569710741239, 3.05040635433004808333399818227, 4.68638538226063907994768758053, 5.84173369858938056260622045457, 6.35867473417400326667114554592, 7.35443250021082381070033471385, 7.959863235009531569939609514323, 8.610267150061525841718405895943, 9.251694058743773098775248300198
