| L(s) = 1 | + (−0.568 − 0.0900i)2-s + (0.349 − 0.685i)3-s + (−1.58 − 0.515i)4-s + (−1.84 + 1.26i)5-s + (−0.260 + 0.358i)6-s + (−1.06 + 0.541i)7-s + (1.88 + 0.959i)8-s + (1.41 + 1.94i)9-s + (1.16 − 0.550i)10-s + (−0.908 + 0.908i)12-s + (0.950 − 6.00i)13-s + (0.653 − 0.212i)14-s + (0.219 + 1.70i)15-s + (1.71 + 1.24i)16-s + (−0.0580 − 0.366i)17-s + (−0.629 − 1.23i)18-s + ⋯ |
| L(s) = 1 | + (−0.402 − 0.0636i)2-s + (0.201 − 0.395i)3-s + (−0.793 − 0.257i)4-s + (−0.825 + 0.563i)5-s + (−0.106 + 0.146i)6-s + (−0.401 + 0.204i)7-s + (0.665 + 0.339i)8-s + (0.471 + 0.649i)9-s + (0.368 − 0.174i)10-s + (−0.262 + 0.262i)12-s + (0.263 − 1.66i)13-s + (0.174 − 0.0567i)14-s + (0.0566 + 0.440i)15-s + (0.428 + 0.311i)16-s + (−0.0140 − 0.0888i)17-s + (−0.148 − 0.291i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.758279 - 0.364028i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.758279 - 0.364028i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.84 - 1.26i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (0.568 + 0.0900i)T + (1.90 + 0.618i)T^{2} \) |
| 3 | \( 1 + (-0.349 + 0.685i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (1.06 - 0.541i)T + (4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (-0.950 + 6.00i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (0.0580 + 0.366i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (0.425 + 1.30i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.48 - 3.48i)T + 23iT^{2} \) |
| 29 | \( 1 + (-1.89 + 5.82i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.00 + 1.45i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.35 + 2.65i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-5.82 + 1.89i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-6.75 + 6.75i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.13 - 0.579i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.402 - 0.0638i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (0.178 + 0.0580i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (0.401 - 0.552i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-7.14 + 7.14i)T - 67iT^{2} \) |
| 71 | \( 1 + (0.836 + 0.607i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.07 - 4.07i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (9.41 - 6.83i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-11.2 + 1.78i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + 8.04iT - 89T^{2} \) |
| 97 | \( 1 + (1.32 - 8.33i)T + (-92.2 - 29.9i)T^{2} \) |
| 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
−10.52822451873349845928889161640, −9.709584875707406247912697727496, −8.656426719049144919065132425480, −7.85624938086210925592855347536, −7.35807877245733483854632359528, −5.99809829438112971891330452741, −4.92621924898538412814746100891, −3.79567956421279283734713004457, −2.59831304327742723171969018366, −0.71174474452233992568196056485,
1.08281068520181658394931764091, 3.39525904277888684946637570803, 4.22720865966237902242559333653, 4.78791062128349640108168271960, 6.52786875481100770838014883940, 7.32745387068757883255758833174, 8.464127632766521547571652366011, 9.015627739020158694452468497826, 9.609676009485745330724599083623, 10.59326192046871220886173836991
