L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 − 0.707i)5-s + (−0.866 + 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.500 + 0.866i)10-s + (1.67 − 0.965i)11-s + (0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (0.965 + 0.258i)20-s + (−1.36 − 1.36i)22-s + 1.00i·25-s + (0.5 − 0.866i)28-s + 0.517·29-s + (−0.866 − 1.5i)31-s + (−0.965 − 0.258i)32-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.707 − 0.707i)5-s + (−0.866 + 0.5i)7-s + (0.707 + 0.707i)8-s + (−0.500 + 0.866i)10-s + (1.67 − 0.965i)11-s + (0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (0.965 + 0.258i)20-s + (−1.36 − 1.36i)22-s + 1.00i·25-s + (0.5 − 0.866i)28-s + 0.517·29-s + (−0.866 − 1.5i)31-s + (−0.965 − 0.258i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6934328129\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6934328129\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
good | 11 | \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - 0.517T + T^{2} \) |
| 31 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
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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
−8.948085183165151248766251265126, −8.396558653818522471894646423011, −7.47721671174138110830149977573, −6.42226043256747039733180225155, −5.59977800936270239682744517229, −4.50676833277190136283862471155, −3.72612506028846570485194313563, −3.19968089701869567744359563511, −1.81886082551746464915034729747, −0.57365088046363720139216528691,
1.32566374558528673505238989977, 3.09679994081083609221431602125, 4.03874719575761070963452762673, 4.48190055880837619006286011237, 5.84959242136761556850422728326, 6.59879831549048116646368293095, 7.10717444479937166745082339146, 7.51597360090906764656848353739, 8.735016055440336813072248356265, 9.169022664902812800555276147153