L(s) = 1 | − 2·4-s − 3.46·5-s − 7-s − 3.46·11-s − 13-s + 4·16-s + 3.46·17-s + 5·19-s + 6.92·20-s + 6.99·25-s + 2·28-s + 31-s + 3.46·35-s + 5·37-s − 3.46·41-s + 8·43-s + 6.92·44-s − 10.3·47-s − 6·49-s + 2·52-s − 3.46·53-s + 11.9·55-s + 13.8·59-s + 5·61-s − 8·64-s + 3.46·65-s + 5·67-s + ⋯ |
L(s) = 1 | − 4-s − 1.54·5-s − 0.377·7-s − 1.04·11-s − 0.277·13-s + 16-s + 0.840·17-s + 1.14·19-s + 1.54·20-s + 1.39·25-s + 0.377·28-s + 0.179·31-s + 0.585·35-s + 0.821·37-s − 0.541·41-s + 1.21·43-s + 1.04·44-s − 1.51·47-s − 0.857·49-s + 0.277·52-s − 0.475·53-s + 1.61·55-s + 1.80·59-s + 0.640·61-s − 64-s + 0.429·65-s + 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6390989594\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6390989594\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 2T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 - 5T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 5T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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
−10.03292564047177048993688046725, −9.463259026207943941620278475260, −8.153523071424416519971803161812, −7.988695506949047678660919363573, −7.01491595581626317194093531625, −5.53415097725298244112775365184, −4.79618401994606835315143031039, −3.78038830838795663436356938691, −3.03372768606895926394732302672, −0.63929328049410835138045782888,
0.63929328049410835138045782888, 3.03372768606895926394732302672, 3.78038830838795663436356938691, 4.79618401994606835315143031039, 5.53415097725298244112775365184, 7.01491595581626317194093531625, 7.988695506949047678660919363573, 8.153523071424416519971803161812, 9.463259026207943941620278475260, 10.03292564047177048993688046725