L(s) = 1 | + (−2.72 − 4.72i)2-s + (−10.8 + 18.8i)4-s + (−2.5 + 4.33i)5-s + (5.90 + 10.2i)7-s + 75.3·8-s + 27.2·10-s + (28.1 + 48.7i)11-s + (−17.2 + 29.9i)13-s + (32.2 − 55.8i)14-s + (−118. − 205. i)16-s + 39.2·17-s − 146.·19-s + (−54.4 − 94.3i)20-s + (153. − 265. i)22-s + (11.7 − 20.4i)23-s + ⋯ |
L(s) = 1 | + (−0.964 − 1.67i)2-s + (−1.36 + 2.35i)4-s + (−0.223 + 0.387i)5-s + (0.318 + 0.552i)7-s + 3.32·8-s + 0.863·10-s + (0.770 + 1.33i)11-s + (−0.369 + 0.639i)13-s + (0.615 − 1.06i)14-s + (−1.84 − 3.20i)16-s + 0.560·17-s − 1.76·19-s + (−0.609 − 1.05i)20-s + (1.48 − 2.57i)22-s + (0.106 − 0.185i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2781190884\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2781190884\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
good | 2 | \( 1 + (2.72 + 4.72i)T + (-4 + 6.92i)T^{2} \) |
| 7 | \( 1 + (-5.90 - 10.2i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-28.1 - 48.7i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (17.2 - 29.9i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 39.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 146.T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-11.7 + 20.4i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (80.5 + 139. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-14.7 + 25.5i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (71.1 - 123. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-234. - 405. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (197. + 341. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 134.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-65.5 + 113. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (129. + 224. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (222. - 385. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 560.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 88.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + (225. + 390. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (142. + 246. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 625.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-96.6 - 167. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.09713299437413921032792824206, −10.15571800033788564991915020001, −9.507437717738665633810627621340, −8.668243053566590786188926949744, −7.78052685605250454627031248035, −6.70728388989306019925589330204, −4.65507733153298550982040049496, −3.86097591586383647691639689788, −2.42568079306610098152788029925, −1.69230472040377580775777739907,
0.14377683630341645679550903822, 1.25637683492598498581851342842, 3.94112664548772255025201716174, 5.14784276610183533018549465027, 6.01421382408648411601889034649, 6.95067779359226052643547568872, 7.86336106245157349060342310806, 8.591780004688504402970639590066, 9.189891056338172441466214433235, 10.42753386513280555205046216723