L(s) = 1 | + (0.0475 + 0.998i)3-s + (−0.327 + 0.945i)4-s + (0.415 + 0.909i)7-s + (−0.995 + 0.0950i)9-s + (−0.959 − 0.281i)12-s + (0.195 + 0.807i)13-s + (−0.786 − 0.618i)16-s + (−0.0947 − 0.00904i)19-s + (−0.888 + 0.458i)21-s + (0.723 − 0.690i)25-s + (−0.142 − 0.989i)27-s + (−0.995 + 0.0950i)28-s + (−0.205 − 0.196i)31-s + (0.235 − 0.971i)36-s + (−0.723 + 1.25i)37-s + ⋯ |
L(s) = 1 | + (0.0475 + 0.998i)3-s + (−0.327 + 0.945i)4-s + (0.415 + 0.909i)7-s + (−0.995 + 0.0950i)9-s + (−0.959 − 0.281i)12-s + (0.195 + 0.807i)13-s + (−0.786 − 0.618i)16-s + (−0.0947 − 0.00904i)19-s + (−0.888 + 0.458i)21-s + (0.723 − 0.690i)25-s + (−0.142 − 0.989i)27-s + (−0.995 + 0.0950i)28-s + (−0.205 − 0.196i)31-s + (0.235 − 0.971i)36-s + (−0.723 + 1.25i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9566036842\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9566036842\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.0475 - 0.998i)T \) |
| 7 | \( 1 + (-0.415 - 0.909i)T \) |
| 67 | \( 1 + (0.959 + 0.281i)T \) |
good | 2 | \( 1 + (0.327 - 0.945i)T^{2} \) |
| 5 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 11 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 13 | \( 1 + (-0.195 - 0.807i)T + (-0.888 + 0.458i)T^{2} \) |
| 17 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (0.0947 + 0.00904i)T + (0.981 + 0.189i)T^{2} \) |
| 23 | \( 1 + (0.995 - 0.0950i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.205 + 0.196i)T + (0.0475 + 0.998i)T^{2} \) |
| 37 | \( 1 + (0.723 - 1.25i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 43 | \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 53 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 59 | \( 1 + (0.888 + 0.458i)T^{2} \) |
| 61 | \( 1 + (-0.0883 - 0.0353i)T + (0.723 + 0.690i)T^{2} \) |
| 71 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 73 | \( 1 + (-0.223 - 1.55i)T + (-0.959 + 0.281i)T^{2} \) |
| 79 | \( 1 + (1.38 + 0.407i)T + (0.841 + 0.540i)T^{2} \) |
| 83 | \( 1 + (-0.723 + 0.690i)T^{2} \) |
| 89 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 97 | \( 1 + (-0.888 - 1.53i)T + (-0.5 + 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.960555008451252892468325161192, −8.961344222118760081662810119996, −8.779013562874237787344218791639, −7.949868505969973447630886766234, −6.85327902207783729038537527626, −5.78295768433281827146572660462, −4.84483905696124795247720365777, −4.20168271455082477117870580128, −3.21027550715460425278467892791, −2.27678197296749974441515285027,
0.821355733680660171654012155922, 1.80907445297401781357238099547, 3.19969047111194341478124646246, 4.45746949348884916709365709926, 5.41298665260482680217402180979, 6.11430731170077706731462987305, 7.06834796277219791911076662929, 7.68109643397127044533236242341, 8.605853885176437001596644652252, 9.328442075177386403392646624250