L(s) = 1 | + (−1.92 − 0.304i)2-s + (2.64 + 0.859i)4-s + (0.649 + 0.760i)5-s + (−3.08 − 1.57i)8-s + (−1.01 − 1.65i)10-s + (0.0922 + 0.126i)11-s + (−0.156 − 0.987i)13-s + (3.20 + 2.32i)16-s + (1.06 + 2.57i)20-s + (−0.138 − 0.271i)22-s + (−0.156 + 0.987i)25-s + 1.94i·26-s + (−2.99 − 2.99i)32-s + (−0.809 − 3.36i)40-s + (1.00 − 1.37i)41-s + ⋯ |
L(s) = 1 | + (−1.92 − 0.304i)2-s + (2.64 + 0.859i)4-s + (0.649 + 0.760i)5-s + (−3.08 − 1.57i)8-s + (−1.01 − 1.65i)10-s + (0.0922 + 0.126i)11-s + (−0.156 − 0.987i)13-s + (3.20 + 2.32i)16-s + (1.06 + 2.57i)20-s + (−0.138 − 0.271i)22-s + (−0.156 + 0.987i)25-s + 1.94i·26-s + (−2.99 − 2.99i)32-s + (−0.809 − 3.36i)40-s + (1.00 − 1.37i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5958520651\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5958520651\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.649 - 0.760i)T \) |
| 13 | \( 1 + (0.156 + 0.987i)T \) |
good | 2 | \( 1 + (1.92 + 0.304i)T + (0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-0.0922 - 0.126i)T + (-0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 29 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (-1.00 + 1.37i)T + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + (-1.39 - 1.39i)T + iT^{2} \) |
| 47 | \( 1 + (-1.51 + 0.774i)T + (0.587 - 0.809i)T^{2} \) |
| 53 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (-1.23 - 0.893i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.734 - 0.533i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 71 | \( 1 + (-1.89 - 0.616i)T + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.416 + 0.211i)T + (0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + (1.49 - 1.08i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.587 - 0.809i)T^{2} \) |
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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
−8.992124291292634363358365374327, −8.399453078526910522881798172903, −7.37806762014532696682894596915, −7.21309783537592853458500601774, −6.15007916424120649198980874495, −5.58188027470757592240757113977, −3.78841073353646610474452506226, −2.75436310478228435322572087729, −2.23123686256784542102187444512, −0.978609565018686973029444541600,
0.920543397487296973921610747175, 1.88063446730073432730790893725, 2.68350486251432594931667655735, 4.28537838330388326663114321076, 5.49690112054397840895117440144, 6.13950948840395998757980069706, 6.85801066568254814597728946275, 7.63840820663507934119315393016, 8.332242804303004179839908400485, 9.122435693068326922326640435578