L(s) = 1 | + (−0.669 + 0.743i)3-s + (−0.207 − 0.978i)4-s + (−0.866 − 0.5i)5-s + (−0.104 − 0.994i)9-s + (−0.994 + 0.104i)11-s + (0.866 + 0.5i)12-s + (0.951 − 0.309i)15-s + (−0.913 + 0.406i)16-s + (−0.309 + 0.951i)20-s + (−0.847 + 0.325i)23-s + (0.499 + 0.866i)25-s + (0.809 + 0.587i)27-s + (−0.544 + 0.604i)31-s + (0.587 − 0.809i)33-s + (−0.951 + 0.309i)36-s + (1.90 − 0.302i)37-s + ⋯ |
L(s) = 1 | + (−0.669 + 0.743i)3-s + (−0.207 − 0.978i)4-s + (−0.866 − 0.5i)5-s + (−0.104 − 0.994i)9-s + (−0.994 + 0.104i)11-s + (0.866 + 0.5i)12-s + (0.951 − 0.309i)15-s + (−0.913 + 0.406i)16-s + (−0.309 + 0.951i)20-s + (−0.847 + 0.325i)23-s + (0.499 + 0.866i)25-s + (0.809 + 0.587i)27-s + (−0.544 + 0.604i)31-s + (0.587 − 0.809i)33-s + (−0.951 + 0.309i)36-s + (1.90 − 0.302i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2533951252\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2533951252\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.994 - 0.104i)T \) |
good | 2 | \( 1 + (0.207 + 0.978i)T^{2} \) |
| 7 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 13 | \( 1 + (-0.207 + 0.978i)T^{2} \) |
| 17 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (0.847 - 0.325i)T + (0.743 - 0.669i)T^{2} \) |
| 29 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 31 | \( 1 + (0.544 - 0.604i)T + (-0.104 - 0.994i)T^{2} \) |
| 37 | \( 1 + (-1.90 + 0.302i)T + (0.951 - 0.309i)T^{2} \) |
| 41 | \( 1 + (-0.978 - 0.207i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (0.0270 - 0.516i)T + (-0.994 - 0.104i)T^{2} \) |
| 53 | \( 1 + (-0.847 - 1.66i)T + (-0.587 + 0.809i)T^{2} \) |
| 59 | \( 1 + (0.0218 - 0.207i)T + (-0.978 - 0.207i)T^{2} \) |
| 61 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 67 | \( 1 + (-0.104 - 1.99i)T + (-0.994 + 0.104i)T^{2} \) |
| 71 | \( 1 + (1.89 + 0.614i)T + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.104 + 0.994i)T^{2} \) |
| 83 | \( 1 + (-0.406 + 0.913i)T^{2} \) |
| 89 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.715 + 0.0375i)T + (0.994 + 0.104i)T^{2} \) |
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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
−9.472198371741145676811366434689, −8.753128592724681169962615608805, −7.87218546876797165104364478757, −7.00933250433081788977217826271, −5.87999094878712285298369878403, −5.49665299440970710394095716439, −4.53920483716466628811019870172, −4.14664527590958946527518610401, −2.83581555328779682040926361751, −1.19710470300721405044675568100,
0.20242239982328667216761797896, 2.21187536239982697292496605813, 3.04119061930110333124920287122, 4.07133358044753852340146859728, 4.85721797441066034641859320634, 5.88332898939834550503193633358, 6.77006688285881323605597718017, 7.42178785822609559486869344039, 8.072015757186355477360058346269, 8.363969849053428075250780603033