| L(s) = 1 | + (0.249 − 0.722i)2-s + (3.86 + 7.49i)3-s + (5.82 + 4.58i)4-s + (0.997 − 0.0952i)5-s + (6.38 − 0.917i)6-s + (9.73 + 15.7i)7-s + (9.91 − 6.36i)8-s + (−25.5 + 35.9i)9-s + (0.180 − 0.744i)10-s + (31.3 − 10.8i)11-s + (−11.8 + 61.4i)12-s + (−23.6 − 80.4i)13-s + (13.8 − 3.09i)14-s + (4.56 + 7.10i)15-s + (11.8 + 48.9i)16-s + (−34.1 − 13.6i)17-s + ⋯ |
| L(s) = 1 | + (0.0883 − 0.255i)2-s + (0.743 + 1.44i)3-s + (0.728 + 0.573i)4-s + (0.0892 − 0.00852i)5-s + (0.434 − 0.0624i)6-s + (0.525 + 0.850i)7-s + (0.438 − 0.281i)8-s + (−0.948 + 1.33i)9-s + (0.00571 − 0.0235i)10-s + (0.859 − 0.297i)11-s + (−0.284 + 1.47i)12-s + (−0.503 − 1.71i)13-s + (0.263 − 0.0590i)14-s + (0.0786 + 0.122i)15-s + (0.185 + 0.764i)16-s + (−0.487 − 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0922 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0922 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.06310 + 1.88079i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06310 + 1.88079i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-9.73 - 15.7i)T \) |
| 23 | \( 1 + (87.4 + 67.1i)T \) |
| good | 2 | \( 1 + (-0.249 + 0.722i)T + (-6.28 - 4.94i)T^{2} \) |
| 3 | \( 1 + (-3.86 - 7.49i)T + (-15.6 + 21.9i)T^{2} \) |
| 5 | \( 1 + (-0.997 + 0.0952i)T + (122. - 23.6i)T^{2} \) |
| 11 | \( 1 + (-31.3 + 10.8i)T + (1.04e3 - 822. i)T^{2} \) |
| 13 | \( 1 + (23.6 + 80.4i)T + (-1.84e3 + 1.18e3i)T^{2} \) |
| 17 | \( 1 + (34.1 + 13.6i)T + (3.55e3 + 3.39e3i)T^{2} \) |
| 19 | \( 1 + (-9.27 + 3.71i)T + (4.96e3 - 4.73e3i)T^{2} \) |
| 29 | \( 1 + (3.51 + 24.4i)T + (-2.34e4 + 6.87e3i)T^{2} \) |
| 31 | \( 1 + (-187. + 8.93i)T + (2.96e4 - 2.83e3i)T^{2} \) |
| 37 | \( 1 + (95.6 + 68.1i)T + (1.65e4 + 4.78e4i)T^{2} \) |
| 41 | \( 1 + (-211. - 96.6i)T + (4.51e4 + 5.20e4i)T^{2} \) |
| 43 | \( 1 + (200. - 311. i)T + (-3.30e4 - 7.23e4i)T^{2} \) |
| 47 | \( 1 + (-437. + 252. i)T + (5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (179. + 188. i)T + (-7.08e3 + 1.48e5i)T^{2} \) |
| 59 | \( 1 + (-554. - 134. i)T + (1.82e5 + 9.41e4i)T^{2} \) |
| 61 | \( 1 + (592. + 305. i)T + (1.31e5 + 1.84e5i)T^{2} \) |
| 67 | \( 1 + (113. + 589. i)T + (-2.79e5 + 1.11e5i)T^{2} \) |
| 71 | \( 1 + (670. + 773. i)T + (-5.09e4 + 3.54e5i)T^{2} \) |
| 73 | \( 1 + (-99.1 + 126. i)T + (-9.17e4 - 3.78e5i)T^{2} \) |
| 79 | \( 1 + (539. - 565. i)T + (-2.34e4 - 4.92e5i)T^{2} \) |
| 83 | \( 1 + (-488. - 1.06e3i)T + (-3.74e5 + 4.32e5i)T^{2} \) |
| 89 | \( 1 + (-29.5 + 620. i)T + (-7.01e5 - 6.70e4i)T^{2} \) |
| 97 | \( 1 + (-360. + 789. i)T + (-5.97e5 - 6.89e5i)T^{2} \) |
| 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
−12.44275201368276307727113184659, −11.54433321661917620493317464732, −10.58618557420411880212225771475, −9.693303352544913656703485023222, −8.571865556053714500478178932267, −7.83239955627109378155205122755, −5.99370421585790752480405227335, −4.61884299983145807228458019579, −3.39357083462486222732867435788, −2.39918849989990557266148305776,
1.38046649086326752506920365655, 2.15731022257596772718110298322, 4.25521724126492846750100790193, 6.17876296480878245259441872438, 7.02481342154480418469219180321, 7.56988586750116894185552153888, 8.886575795073330588153494876026, 10.11337190583088221968124087922, 11.58735791178587412592161144687, 12.00908431436539208668842138788
