L(s) = 1 | − 4·11-s − 13-s + 19-s − 4·23-s − 3·29-s − 4·31-s − 5·37-s − 9·41-s − 2·43-s + 3·47-s − 7·49-s − 53-s − 10·59-s − 4·61-s + 9·67-s + 7·71-s − 4·73-s − 11·79-s + 6·83-s − 10·89-s + 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 1.20·11-s − 0.277·13-s + 0.229·19-s − 0.834·23-s − 0.557·29-s − 0.718·31-s − 0.821·37-s − 1.40·41-s − 0.304·43-s + 0.437·47-s − 49-s − 0.137·53-s − 1.30·59-s − 0.512·61-s + 1.09·67-s + 0.830·71-s − 0.468·73-s − 1.23·79-s + 0.658·83-s − 1.05·89-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.08913420048\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08913420048\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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
−13.02328111336642, −12.74450456277215, −12.20965260896005, −11.74182365414373, −11.22284937977053, −10.74247708958853, −10.28585831479891, −9.873140348167526, −9.424398715605611, −8.763252264594476, −8.360674656175340, −7.760080620964669, −7.503354788588313, −6.866843199856722, −6.357142623251004, −5.694895837477172, −5.310148467927146, −4.833321663697894, −4.272187003426179, −3.442184284721482, −3.234474874258857, −2.358440581071075, −1.945064620222947, −1.245607837973191, −0.08304301364518524,
0.08304301364518524, 1.245607837973191, 1.945064620222947, 2.358440581071075, 3.234474874258857, 3.442184284721482, 4.272187003426179, 4.833321663697894, 5.310148467927146, 5.694895837477172, 6.357142623251004, 6.866843199856722, 7.503354788588313, 7.760080620964669, 8.360674656175340, 8.763252264594476, 9.424398715605611, 9.873140348167526, 10.28585831479891, 10.74247708958853, 11.22284937977053, 11.74182365414373, 12.20965260896005, 12.74450456277215, 13.02328111336642