L(s) = 1 | + (−1.17 + 0.780i)2-s + (1.26 + 1.17i)3-s + (0.780 − 1.84i)4-s − 0.662·5-s + (−2.41 − 0.399i)6-s + 3.83·7-s + (0.516 + 2.78i)8-s + (0.219 + 2.99i)9-s + (0.780 − 0.516i)10-s − 2i·11-s + (3.16 − 1.41i)12-s + (−2.56 + 2.53i)13-s + (−4.51 + 2.99i)14-s + (−0.840 − 0.780i)15-s + (−2.78 − 2.87i)16-s + 1.68i·17-s + ⋯ |
L(s) = 1 | + (−0.833 + 0.552i)2-s + (0.732 + 0.680i)3-s + (0.390 − 0.920i)4-s − 0.296·5-s + (−0.986 − 0.163i)6-s + 1.44·7-s + (0.182 + 0.983i)8-s + (0.0730 + 0.997i)9-s + (0.246 − 0.163i)10-s − 0.603i·11-s + (0.912 − 0.408i)12-s + (−0.710 + 0.703i)13-s + (−1.20 + 0.799i)14-s + (−0.216 − 0.201i)15-s + (−0.695 − 0.718i)16-s + 0.407i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.837514 + 0.573944i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.837514 + 0.573944i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.17 - 0.780i)T \) |
| 3 | \( 1 + (-1.26 - 1.17i)T \) |
| 13 | \( 1 + (2.56 - 2.53i)T \) |
good | 5 | \( 1 + 0.662T + 5T^{2} \) |
| 7 | \( 1 - 3.83T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 17 | \( 1 - 1.68iT - 17T^{2} \) |
| 19 | \( 1 + 2.15T + 19T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 + 7.66iT - 29T^{2} \) |
| 31 | \( 1 - 2.15T + 31T^{2} \) |
| 37 | \( 1 + 9.03iT - 37T^{2} \) |
| 41 | \( 1 + 6.41T + 41T^{2} \) |
| 43 | \( 1 + 3.68iT - 43T^{2} \) |
| 47 | \( 1 + 6.68iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 2.87iT - 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + 5.51T + 67T^{2} \) |
| 71 | \( 1 - 14.6iT - 71T^{2} \) |
| 73 | \( 1 - 12.9iT - 73T^{2} \) |
| 79 | \( 1 + 8.10iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 7.92iT - 97T^{2} \) |
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
−13.58884530935901989229595882313, −11.64412898556199432607744519970, −10.96258801848474190001620391652, −9.917528634078574513212436865893, −8.788561774988112750346942548907, −8.174254830949846867152677079668, −7.20884394960871195603116752378, −5.44042494440734481690143367974, −4.30503593896860276149873350559, −2.11779064044379284507143985945,
1.54816616362882618813912935008, 2.96660041878771859587686966245, 4.70995968590018575335526481723, 6.95509127531797161846434679514, 7.78693085779490438738468302857, 8.468529798679586624977887690997, 9.560732810505596304516148841439, 10.77084606227869786926871523994, 11.78437783135770655849669764751, 12.50613171645055754958744517230