L(s) = 1 | + (−2.97 − 2.97i)2-s + (−1.22 + 1.22i)3-s + 1.69i·4-s + (−22.1 + 11.5i)5-s + 7.31·6-s + (13.0 + 13.0i)7-s + (−42.5 + 42.5i)8-s + 77.9i·9-s + (100. + 31.6i)10-s + 17.6·11-s + (−2.07 − 2.07i)12-s + (−160. + 160. i)13-s − 77.8i·14-s + (13.0 − 41.4i)15-s + 280.·16-s + (−324. − 324. i)17-s + ⋯ |
L(s) = 1 | + (−0.743 − 0.743i)2-s + (−0.136 + 0.136i)3-s + 0.105i·4-s + (−0.887 + 0.461i)5-s + 0.203·6-s + (0.267 + 0.267i)7-s + (−0.664 + 0.664i)8-s + 0.962i·9-s + (1.00 + 0.316i)10-s + 0.145·11-s + (−0.0144 − 0.0144i)12-s + (−0.949 + 0.949i)13-s − 0.397i·14-s + (0.0581 − 0.184i)15-s + 1.09·16-s + (−1.12 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.171381 + 0.220213i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.171381 + 0.220213i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (22.1 - 11.5i)T \) |
| 7 | \( 1 + (-13.0 - 13.0i)T \) |
good | 2 | \( 1 + (2.97 + 2.97i)T + 16iT^{2} \) |
| 3 | \( 1 + (1.22 - 1.22i)T - 81iT^{2} \) |
| 11 | \( 1 - 17.6T + 1.46e4T^{2} \) |
| 13 | \( 1 + (160. - 160. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (324. + 324. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 + 468. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (625. - 625. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 - 755. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 553.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-555. - 555. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.68e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (416. - 416. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (319. + 319. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-3.35e3 + 3.35e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 4.67e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 848.T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-2.47e3 - 2.47e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 2.45e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (2.34e3 - 2.34e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 - 1.76e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (885. - 885. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 3.51e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-421. - 421. i)T + 8.85e7iT^{2} \) |
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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
−16.15218824505924921314590598497, −15.05938413252424937518409238427, −13.78857513995934216651207045555, −11.69713259332858045782986928164, −11.36243225777173653041993399546, −9.979000796722858180062149636127, −8.679082741069799225717265600368, −7.14565299041767772032382781919, −4.83747352752917552830315097755, −2.39684078814528806036840330777,
0.24389848267706104084855796792, 3.97648084660056758964217946186, 6.30450104464202513401945755444, 7.74963407457418344147490351055, 8.580928546412894993593974202187, 10.12639389228431795656759355629, 11.98588142666724247474194832048, 12.69606272981120217779095191212, 14.80844390128628729232829843656, 15.55626288145457611131856494741