| L(s) = 1 | + (0.756 + 0.0304i)2-s + (0.278 − 0.960i)3-s + (−1.42 − 0.114i)4-s + (−1.80 − 0.683i)5-s + (0.239 − 0.717i)6-s + (0.988 + 2.32i)7-s + (−2.57 − 0.312i)8-s + (−0.845 − 0.534i)9-s + (−1.34 − 0.571i)10-s + (−3.17 − 5.02i)11-s + (−0.506 + 1.33i)12-s + (0.984 + 3.46i)13-s + (0.676 + 1.78i)14-s + (−1.15 + 1.54i)15-s + (0.879 + 0.142i)16-s + (−6.63 + 2.82i)17-s + ⋯ |
| L(s) = 1 | + (0.534 + 0.0215i)2-s + (0.160 − 0.554i)3-s + (−0.711 − 0.0574i)4-s + (−0.806 − 0.305i)5-s + (0.0978 − 0.293i)6-s + (0.373 + 0.877i)7-s + (−0.910 − 0.110i)8-s + (−0.281 − 0.178i)9-s + (−0.424 − 0.180i)10-s + (−0.957 − 1.51i)11-s + (−0.146 + 0.385i)12-s + (0.272 + 0.962i)13-s + (0.180 + 0.477i)14-s + (−0.299 + 0.397i)15-s + (0.219 + 0.0357i)16-s + (−1.60 + 0.685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0175299 + 0.177539i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0175299 + 0.177539i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.278 + 0.960i)T \) |
| 13 | \( 1 + (-0.984 - 3.46i)T \) |
| good | 2 | \( 1 + (-0.756 - 0.0304i)T + (1.99 + 0.160i)T^{2} \) |
| 5 | \( 1 + (1.80 + 0.683i)T + (3.74 + 3.31i)T^{2} \) |
| 7 | \( 1 + (-0.988 - 2.32i)T + (-4.84 + 5.04i)T^{2} \) |
| 11 | \( 1 + (3.17 + 5.02i)T + (-4.71 + 9.93i)T^{2} \) |
| 17 | \( 1 + (6.63 - 2.82i)T + (11.7 - 12.2i)T^{2} \) |
| 19 | \( 1 + (4.97 - 2.87i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.77 + 4.80i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.236 - 5.88i)T + (-28.9 - 2.33i)T^{2} \) |
| 31 | \( 1 + (6.21 + 7.01i)T + (-3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-5.74 + 1.17i)T + (34.0 - 14.5i)T^{2} \) |
| 41 | \( 1 + (3.28 + 0.952i)T + (34.6 + 21.9i)T^{2} \) |
| 43 | \( 1 + (1.59 - 7.81i)T + (-39.5 - 16.8i)T^{2} \) |
| 47 | \( 1 + (-4.44 + 3.06i)T + (16.6 - 43.9i)T^{2} \) |
| 53 | \( 1 + (-1.15 + 9.51i)T + (-51.4 - 12.6i)T^{2} \) |
| 59 | \( 1 + (-0.547 - 3.36i)T + (-55.9 + 18.6i)T^{2} \) |
| 61 | \( 1 + (-0.266 + 0.200i)T + (16.9 - 58.5i)T^{2} \) |
| 67 | \( 1 + (0.813 + 10.0i)T + (-66.1 + 10.7i)T^{2} \) |
| 71 | \( 1 + (2.56 - 2.45i)T + (2.85 - 70.9i)T^{2} \) |
| 73 | \( 1 + (-2.44 + 4.66i)T + (-41.4 - 60.0i)T^{2} \) |
| 79 | \( 1 + (-0.769 - 1.11i)T + (-28.0 + 73.8i)T^{2} \) |
| 83 | \( 1 + (-2.02 + 8.22i)T + (-73.4 - 38.5i)T^{2} \) |
| 89 | \( 1 + (0.207 + 0.119i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.273 - 0.223i)T + (19.4 + 95.0i)T^{2} \) |
| 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
−10.79221887989676470701664068701, −9.074076010840784960646659668506, −8.552613853927231593080260532592, −8.112685732038763118712233293480, −6.49293935734024305138485234987, −5.74699623103986840813274496301, −4.60896520620336448287784330022, −3.71147265805988728865365238259, −2.29059633348079707807446120453, −0.084490387659648233950476851658,
2.66834984423809495833787480111, 3.96746351322501943355513034341, 4.52438194427415593984981409237, 5.35543709801094670624079535532, 7.01879204203397788210898812350, 7.73185993062263267675894967589, 8.722249991174428553261800183499, 9.657594796586141865092306641986, 10.63781793440920214401238804690, 11.19011793720383759945575875819
