L(s) = 1 | − i·2-s − 3-s + 4-s + i·6-s + 2i·7-s − 3i·8-s + 9-s − 12-s + (3 + 2i)13-s + 2·14-s − 16-s − 2·17-s − i·18-s + 2i·19-s − 2i·21-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.577·3-s + 0.5·4-s + 0.408i·6-s + 0.755i·7-s − 1.06i·8-s + 0.333·9-s − 0.288·12-s + (0.832 + 0.554i)13-s + 0.534·14-s − 0.250·16-s − 0.485·17-s − 0.235i·18-s + 0.458i·19-s − 0.436i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58517 - 0.479953i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58517 - 0.479953i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3 - 2i)T \) |
good | 2 | \( 1 + iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 6iT - 67T^{2} \) |
| 71 | \( 1 - 8iT - 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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
−10.13336608059775977039909759168, −9.250026521604170608471763914047, −8.449525282304171456462960788926, −7.18130209047634196329054430817, −6.48132365688038266589261786180, −5.70165687914197328129668958458, −4.55238946818830063525665263081, −3.42702303067957285369968240427, −2.34710161549458293039409910544, −1.17492805803391975860218363320,
1.04007413228782745627927839378, 2.66277753723284712249708639345, 3.98059232606463378037360849824, 5.12868739012297732474610252066, 5.85380573295926455919206720937, 6.88422917842219885207653566343, 7.18786716813904239023266219348, 8.276703869195701940548040106465, 9.073911143958808428783703376788, 10.37175477252866260186786002138