| L(s) = 1 | + (0.173 − 0.984i)2-s + (−1.70 − 0.300i)3-s + (−0.939 − 0.342i)4-s + (−1.03 + 0.866i)5-s + (−0.592 + 1.62i)6-s + (−0.939 + 0.342i)7-s + (−0.5 + 0.866i)8-s + (2.81 + 1.02i)9-s + (0.673 + 1.16i)10-s + (3.64 + 3.05i)11-s + (1.49 + 0.866i)12-s + (0.266 + 1.50i)13-s + (0.173 + 0.984i)14-s + (2.02 − 1.16i)15-s + (0.766 + 0.642i)16-s + (−1.11 − 1.92i)17-s + ⋯ |
| L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.984 − 0.173i)3-s + (−0.469 − 0.171i)4-s + (−0.461 + 0.387i)5-s + (−0.241 + 0.664i)6-s + (−0.355 + 0.129i)7-s + (−0.176 + 0.306i)8-s + (0.939 + 0.342i)9-s + (0.213 + 0.368i)10-s + (1.09 + 0.922i)11-s + (0.433 + 0.249i)12-s + (0.0737 + 0.418i)13-s + (0.0464 + 0.263i)14-s + (0.521 − 0.301i)15-s + (0.191 + 0.160i)16-s + (−0.270 − 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.871721 + 0.0507719i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.871721 + 0.0507719i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (1.70 + 0.300i)T \) |
| 7 | \( 1 + (0.939 - 0.342i)T \) |
| good | 5 | \( 1 + (1.03 - 0.866i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (-3.64 - 3.05i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.266 - 1.50i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.11 + 1.92i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.79 + 4.84i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.57 - 2.75i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.85 - 10.5i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.37 - 0.502i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.815 - 1.41i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.75 - 9.97i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (6.70 + 5.63i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-8.35 + 3.03i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 1.32T + 53T^{2} \) |
| 59 | \( 1 + (7.16 - 6.01i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.08 + 1.12i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.470 + 2.66i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.10 + 3.64i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.54 - 7.87i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.556 + 3.15i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (2.22 - 12.6i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (0.779 - 1.35i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.91 + 8.31i)T + (16.8 + 95.5i)T^{2} \) |
| 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
−11.42910366964784561219462478145, −10.82561694135699951935131934885, −9.624311882683919499542113070624, −9.028039381733479485965923485953, −7.10996443877423341199700773964, −6.89997068848061516252607600840, −5.33058048778540071897575458397, −4.44483479195359510960871454432, −3.16563681769817311945100985196, −1.36587140397128264326838243014,
0.75551355626911789136133263968, 3.63334277362969873672709770248, 4.48056608614308113383340474784, 5.77420506695725453505405418021, 6.33094094610710624952731851290, 7.44727101291300540171248202030, 8.496728257358985066141139419258, 9.454610451456002526289722313124, 10.47502985172644888923077277655, 11.47620145947200787826983742815
