| L(s) = 1 | + (−0.190 − 0.587i)2-s + (−0.827 − 0.918i)3-s + (1.30 − 0.951i)4-s + (−0.5 − 0.866i)5-s + (−0.381 + 0.661i)6-s + (0.442 + 4.21i)7-s + (−1.80 − 1.31i)8-s + (0.153 − 1.46i)9-s + (−0.413 + 0.459i)10-s + (−1.82 − 0.813i)11-s + (−1.95 − 0.415i)12-s + (1.20 − 0.256i)13-s + (2.39 − 1.06i)14-s + (−0.381 + 1.17i)15-s + (0.572 − 1.76i)16-s + (−4.78 + 2.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.135 − 0.415i)2-s + (−0.477 − 0.530i)3-s + (0.654 − 0.475i)4-s + (−0.223 − 0.387i)5-s + (−0.155 + 0.270i)6-s + (0.167 + 1.59i)7-s + (−0.639 − 0.464i)8-s + (0.0512 − 0.488i)9-s + (−0.130 + 0.145i)10-s + (−0.550 − 0.245i)11-s + (−0.564 − 0.120i)12-s + (0.335 − 0.0712i)13-s + (0.639 − 0.284i)14-s + (−0.0986 + 0.303i)15-s + (0.143 − 0.440i)16-s + (−1.16 + 0.516i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0910887 + 0.659289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0910887 + 0.659289i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 \) |
| good | 2 | \( 1 + (0.190 + 0.587i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (0.827 + 0.918i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.442 - 4.21i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (1.82 + 0.813i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-1.20 + 0.256i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (4.78 - 2.12i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (2.18 + 0.464i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (6.23 + 4.53i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.23 + 6.88i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.68 + 5.20i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (3.16 + 0.672i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (2 - 6.15i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.159 - 1.51i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (1.49 + 1.66i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.37 - 13.1i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-0.431 - 0.192i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (1.56 - 0.694i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (1.97 - 2.18i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (1.38 - 1.00i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.57 - 1.14i)T + (29.9 - 92.2i)T^{2} \) |
| 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
−9.592161464655201244945993968362, −8.725305721962600965083384038940, −8.109560237099609860537541915349, −6.69324967946663938881303874793, −6.12031037688730516643622789958, −5.54276698167554388088930707762, −4.19181902599993387236173495526, −2.63269514415010715267135077926, −1.91424092126641772955543096281, −0.31011149207910842362027125560,
1.97254207134369230581726673355, 3.42648652328988362792831464213, 4.26691811421763502090819724512, 5.26528358452951392339246721813, 6.42136981842489987524430220854, 7.25918718998078888649060343386, 7.64150371016226345740427746062, 8.623523511879638682963578031188, 9.897511711647613847134699482697, 10.65015927224584124648674464305
