| L(s) = 1 | − 2.23i·5-s − 9.55i·7-s + 18.8i·11-s + 20.1·13-s − 28.4i·17-s + 28.9i·19-s + (18.3 + 13.8i)23-s − 5.00·25-s − 1.59·29-s − 26.9·31-s − 21.3·35-s + 51.7i·37-s + 54.4·41-s + 81.2i·43-s − 27.5·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s − 1.36i·7-s + 1.70i·11-s + 1.54·13-s − 1.67i·17-s + 1.52i·19-s + (0.799 + 0.600i)23-s − 0.200·25-s − 0.0548·29-s − 0.867·31-s − 0.610·35-s + 1.39i·37-s + 1.32·41-s + 1.89i·43-s − 0.586·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.910052349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910052349\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-18.3 - 13.8i)T \) |
| good | 7 | \( 1 + 9.55iT - 49T^{2} \) |
| 11 | \( 1 - 18.8iT - 121T^{2} \) |
| 13 | \( 1 - 20.1T + 169T^{2} \) |
| 17 | \( 1 + 28.4iT - 289T^{2} \) |
| 19 | \( 1 - 28.9iT - 361T^{2} \) |
| 29 | \( 1 + 1.59T + 841T^{2} \) |
| 31 | \( 1 + 26.9T + 961T^{2} \) |
| 37 | \( 1 - 51.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 54.4T + 1.68e3T^{2} \) |
| 43 | \( 1 - 81.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 27.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 37.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 53.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 39.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 42.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 84.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 77.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 80.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 119. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 22.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 99.9iT - 9.40e3T^{2} \) |
| 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
−8.170701138145764218305774970014, −7.58682007781384647136067989564, −7.03293449331656150433422233263, −6.25831801979565226287022701460, −5.24136257602425254404507717767, −4.50807140968098534944167619108, −3.92086859048183934458890859015, −3.02042927994240449299602150242, −1.57422158494988204568215050181, −1.06825237062773246638072560822,
0.43146342025896267430032993802, 1.68533074246324044657786646949, 2.73887334160511781574521755525, 3.37579711111507805044521668921, 4.21169392296978128654410926447, 5.53315484307755127822002148328, 5.90778498727446340497410227060, 6.40628727905898071845926099711, 7.43669436042665931508848875274, 8.434073771088134806055500672956
