L(s) = 1 | + 3·5-s − 7-s − 3·11-s − 17-s + 19-s − 7·23-s + 4·25-s + 5·29-s − 6·31-s − 3·35-s + 3·37-s − 2·41-s + 43-s + 49-s − 2·53-s − 9·55-s + 10·59-s + 11·61-s − 8·67-s + 2·71-s + 7·73-s + 3·77-s + 10·79-s + 14·83-s − 3·85-s − 2·89-s + 3·95-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 0.377·7-s − 0.904·11-s − 0.242·17-s + 0.229·19-s − 1.45·23-s + 4/5·25-s + 0.928·29-s − 1.07·31-s − 0.507·35-s + 0.493·37-s − 0.312·41-s + 0.152·43-s + 1/7·49-s − 0.274·53-s − 1.21·55-s + 1.30·59-s + 1.40·61-s − 0.977·67-s + 0.237·71-s + 0.819·73-s + 0.341·77-s + 1.12·79-s + 1.53·83-s − 0.325·85-s − 0.211·89-s + 0.307·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−14.13953896297697, −13.55757761295488, −13.26514126910409, −12.88486313269670, −12.19836265073858, −11.84646470923645, −11.05164267335936, −10.54305035009121, −10.19136751985083, −9.695889885511585, −9.313303792682402, −8.739105705596266, −7.998884620403203, −7.760426474316571, −6.792885917639906, −6.535543990070068, −5.892716323742071, −5.392438670652418, −5.070647935817638, −4.177147469215715, −3.628458228852332, −2.793728147508169, −2.303146067550825, −1.843805089578930, −0.9288437439172183, 0,
0.9288437439172183, 1.843805089578930, 2.303146067550825, 2.793728147508169, 3.628458228852332, 4.177147469215715, 5.070647935817638, 5.392438670652418, 5.892716323742071, 6.535543990070068, 6.792885917639906, 7.760426474316571, 7.998884620403203, 8.739105705596266, 9.313303792682402, 9.695889885511585, 10.19136751985083, 10.54305035009121, 11.05164267335936, 11.84646470923645, 12.19836265073858, 12.88486313269670, 13.26514126910409, 13.55757761295488, 14.13953896297697