L(s) = 1 | + 2-s + (0.184 − 1.72i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.184 − 1.72i)6-s + (−2.25 − 1.38i)7-s + 8-s + (−2.93 − 0.636i)9-s + (−0.5 + 0.866i)10-s + (−0.480 − 0.832i)11-s + (0.184 − 1.72i)12-s + (−2.94 − 5.09i)13-s + (−2.25 − 1.38i)14-s + (1.39 + 1.02i)15-s + 16-s + (3.89 − 6.75i)17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.106 − 0.994i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.0754 − 0.703i)6-s + (−0.853 − 0.521i)7-s + 0.353·8-s + (−0.977 − 0.212i)9-s + (−0.158 + 0.273i)10-s + (−0.144 − 0.251i)11-s + (0.0533 − 0.497i)12-s + (−0.816 − 1.41i)13-s + (−0.603 − 0.368i)14-s + (0.361 + 0.263i)15-s + 0.250·16-s + (0.945 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.904661 - 1.48581i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.904661 - 1.48581i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.184 + 1.72i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.25 + 1.38i)T \) |
good | 11 | \( 1 + (0.480 + 0.832i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.94 + 5.09i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.89 + 6.75i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.774 - 1.34i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.95 + 3.38i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.543 - 0.940i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.55T + 31T^{2} \) |
| 37 | \( 1 + (-5.84 - 10.1i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.15 - 1.99i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.18 - 3.79i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.55T + 47T^{2} \) |
| 53 | \( 1 + (-0.274 + 0.475i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 6.00T + 59T^{2} \) |
| 61 | \( 1 - 14.6T + 61T^{2} \) |
| 67 | \( 1 + 0.821T + 67T^{2} \) |
| 71 | \( 1 + 3.96T + 71T^{2} \) |
| 73 | \( 1 + (-0.368 + 0.638i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 6.79T + 79T^{2} \) |
| 83 | \( 1 + (-6.12 + 10.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.85 + 4.94i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.80 + 13.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.35801868914504383203637964491, −9.653930774845072064264119746218, −8.140582867277995561772444275055, −7.48580595442995166569341231644, −6.77654474824801063841277885004, −5.86771560875303269490697830828, −4.86455928277494741533686672605, −3.10568321747246964990529782146, −2.89078620225033700797945586114, −0.73178261100985527070782302072,
2.24892752690252664829405138641, 3.52200442462765191702375157145, 4.26266306132629945061625317403, 5.30122452454696558340248579281, 6.07703825585450740101748403761, 7.23162613817051203330405182775, 8.396875785536013588023278056024, 9.369116263076361730700362138063, 9.886823750268110180003826858988, 10.93376734295419343515533498893