| L(s) = 1 | − 1.41·2-s + 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (2.03 + 6.69i)7-s − 2.82·8-s − 2.99·9-s − 5.30·11-s + 3.46i·12-s + 15.0i·13-s + (−2.87 − 9.47i)14-s + 4.00·16-s − 1.80i·17-s + 4.24·18-s + 7.35i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (0.290 + 0.956i)7-s − 0.353·8-s − 0.333·9-s − 0.482·11-s + 0.288i·12-s + 1.15i·13-s + (−0.205 − 0.676i)14-s + 0.250·16-s − 0.106i·17-s + 0.235·18-s + 0.387i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5590084905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5590084905\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.03 - 6.69i)T \) |
| good | 11 | \( 1 + 5.30T + 121T^{2} \) |
| 13 | \( 1 - 15.0iT - 169T^{2} \) |
| 17 | \( 1 + 1.80iT - 289T^{2} \) |
| 19 | \( 1 - 7.35iT - 361T^{2} \) |
| 23 | \( 1 - 15.8T + 529T^{2} \) |
| 29 | \( 1 + 44.3T + 841T^{2} \) |
| 31 | \( 1 - 6.10iT - 961T^{2} \) |
| 37 | \( 1 + 36.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 18.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 36.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 26.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 80.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 38.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 25.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 90.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 41.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 106. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 16.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 34.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 58.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 140. iT - 9.40e3T^{2} \) |
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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
−10.06612943108379955295110505759, −9.111355559849256282719491620980, −8.887085579527511314997518129083, −7.82778351253703083315235015257, −6.95176159126061071617485489842, −5.85777554239624417661247940040, −5.13048611947366476841522871187, −3.95590017485363931941324429362, −2.71933218446417870395795410742, −1.70814939706610615474300807493,
0.22387147196429966361314943978, 1.29124139684506426275878065578, 2.58751277770053974273658983556, 3.69683712222567315662779795933, 5.08432070958375691146623723405, 5.96932634787640990141126552461, 7.17294523234821271190384165098, 7.49938909932051684387127521341, 8.341823178436091676859666641910, 9.171170206261173720798351035604
