L(s) = 1 | + (0.309 − 0.951i)5-s + (−0.809 + 0.587i)9-s + (0.5 − 0.363i)13-s + (1.11 + 0.363i)17-s + (−0.809 − 0.587i)25-s + (1.11 − 0.363i)29-s + (1.30 − 0.951i)37-s + (0.5 − 0.363i)41-s + (0.309 + 0.951i)45-s − 49-s + (−0.190 − 0.587i)53-s + (1.11 − 1.53i)61-s + (−0.190 − 0.587i)65-s + (−1.11 + 1.53i)73-s + (0.309 − 0.951i)81-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)5-s + (−0.809 + 0.587i)9-s + (0.5 − 0.363i)13-s + (1.11 + 0.363i)17-s + (−0.809 − 0.587i)25-s + (1.11 − 0.363i)29-s + (1.30 − 0.951i)37-s + (0.5 − 0.363i)41-s + (0.309 + 0.951i)45-s − 49-s + (−0.190 − 0.587i)53-s + (1.11 − 1.53i)61-s + (−0.190 − 0.587i)65-s + (−1.11 + 1.53i)73-s + (0.309 − 0.951i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.305904375\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.305904375\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
good | 3 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \) |
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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
−8.494004253786091339328075903577, −8.266116756572131898991523211592, −7.47313481979949951285097830760, −6.23838537211485921702912498203, −5.70150332984972692526655503962, −5.04172528733678834429827788973, −4.15702597334876755151653822199, −3.13350887587867783382647697112, −2.11710160794045514991410763676, −0.926360685319392811100748168732,
1.26244889228972097697139828913, 2.70439904790741946822667987250, 3.16348663325660160583943718081, 4.15855858873305867655114064649, 5.27383778758679779649587760825, 6.10765773794157223890110576121, 6.51104193069575031147074039316, 7.43526818493352817967250256756, 8.163473977534815592573104105751, 8.975520498564265504056300196794