L(s) = 1 | + (−2.71 + 1.56i)3-s + (−1.5 + 0.866i)5-s + 2i·7-s + (3.40 − 5.90i)9-s + (0.225 − 0.391i)13-s + (2.71 − 4.69i)15-s + (−2.90 − 5.03i)17-s + (−4.16 + 1.27i)19-s + (−3.13 − 5.42i)21-s + (−5.70 − 3.29i)23-s + (−1 + 1.73i)25-s + 11.9i·27-s + (2.90 − 5.03i)29-s + 8.33·31-s + (−1.73 − 3i)35-s + ⋯ |
L(s) = 1 | + (−1.56 + 0.904i)3-s + (−0.670 + 0.387i)5-s + 0.755i·7-s + (1.13 − 1.96i)9-s + (0.0626 − 0.108i)13-s + (0.700 − 1.21i)15-s + (−0.704 − 1.22i)17-s + (−0.956 + 0.292i)19-s + (−0.683 − 1.18i)21-s + (−1.18 − 0.686i)23-s + (−0.200 + 0.346i)25-s + 2.29i·27-s + (0.539 − 0.934i)29-s + 1.49·31-s + (−0.292 − 0.507i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4890474210\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4890474210\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (4.16 - 1.27i)T \) |
good | 3 | \( 1 + (2.71 - 1.56i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + (-0.225 + 0.391i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.90 + 5.03i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (5.70 + 3.29i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.90 + 5.03i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.33T + 31T^{2} \) |
| 37 | \( 1 + 3.45T + 37T^{2} \) |
| 41 | \( 1 + (0.677 - 0.391i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.276 + 0.478i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.13 - 4.69i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.31 + 7.47i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.506 + 0.292i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.8 - 6.29i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12.5 + 7.24i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.91 - 8.52i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.18 + 5.52i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8.33T + 83T^{2} \) |
| 89 | \( 1 + (9.11 + 5.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.677 - 0.391i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.02294210872463674965850429809, −9.072172525596704239864773229145, −8.149308262097960205727999714114, −6.93649662560297627475192378500, −6.25506653197601154348228824959, −5.51256678773750978783564857619, −4.55235783218757804914923990731, −3.96747922519301113378378953301, −2.53727020270266348833416374544, −0.38148836422816017410721538379,
0.801818992563391391980865125158, 1.99132032170354901986793864279, 4.00948465060556961765980665327, 4.57679640233986540894148321551, 5.70284824981965446110165499149, 6.45964813033117549893488493855, 7.07753015900573235879889416532, 7.968331393613399421408364766006, 8.629264420181450057614579713859, 10.21340868651861116654865878070