| L(s) = 1 | − 2.61i·3-s − 3.96i·5-s − 7-s − 3.84·9-s − 0.474i·11-s + 1.64i·13-s − 10.3·15-s − 1.49·17-s − 6.98i·19-s + 2.61i·21-s + 1.46·23-s − 10.6·25-s + 2.20i·27-s − 6.47i·29-s − 4.57·31-s + ⋯ |
| L(s) = 1 | − 1.51i·3-s − 1.77i·5-s − 0.377·7-s − 1.28·9-s − 0.143i·11-s + 0.455i·13-s − 2.67·15-s − 0.363·17-s − 1.60i·19-s + 0.570i·21-s + 0.305·23-s − 2.13·25-s + 0.424i·27-s − 1.20i·29-s − 0.820·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8991271677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8991271677\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 2.61iT - 3T^{2} \) |
| 5 | \( 1 + 3.96iT - 5T^{2} \) |
| 11 | \( 1 + 0.474iT - 11T^{2} \) |
| 13 | \( 1 - 1.64iT - 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 + 6.98iT - 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 6.47iT - 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 - 8.73iT - 37T^{2} \) |
| 41 | \( 1 - 7.51T + 41T^{2} \) |
| 43 | \( 1 - 11.8iT - 43T^{2} \) |
| 47 | \( 1 + 5.34T + 47T^{2} \) |
| 53 | \( 1 - 11.8iT - 53T^{2} \) |
| 59 | \( 1 + 11.6iT - 59T^{2} \) |
| 61 | \( 1 + 7.76iT - 61T^{2} \) |
| 67 | \( 1 + 1.19iT - 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 - 7.82T + 89T^{2} \) |
| 97 | \( 1 - 9.03T + 97T^{2} \) |
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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
−7.909069576556645602277832954009, −7.41659371969697170899139024099, −6.41905376780017606253592363057, −6.01158922101239221936398021829, −4.86658798784713242040212045288, −4.42806889161471122510009324723, −2.97938639274215891130180490664, −1.96586050558542126425201059403, −1.11565486065146754945531454274, −0.28291581068395822739945260398,
2.10109385304774756329625641730, 3.14181315507267242907708123032, 3.59115232728718276679549399898, 4.26132882631934456885679878605, 5.52732144646050836967928354999, 5.88949506703417929543314065759, 7.00642757096156983093154586254, 7.42501961464688435379328547595, 8.561080284952030649966132743066, 9.282113620658636380997083465232
