L(s) = 1 | + (−0.809 + 0.587i)3-s − 0.618·7-s + (0.587 + 0.190i)13-s + (0.951 − 1.30i)19-s + (0.500 − 0.363i)21-s + (0.309 + 0.951i)23-s + (−0.309 − 0.951i)27-s + (1.30 − 0.951i)29-s + (−0.587 + 0.809i)31-s + (0.951 + 0.309i)37-s + (−0.587 + 0.190i)39-s − 1.61·43-s + (−0.5 + 0.363i)47-s − 0.618·49-s + (0.587 + 0.809i)53-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)3-s − 0.618·7-s + (0.587 + 0.190i)13-s + (0.951 − 1.30i)19-s + (0.500 − 0.363i)21-s + (0.309 + 0.951i)23-s + (−0.309 − 0.951i)27-s + (1.30 − 0.951i)29-s + (−0.587 + 0.809i)31-s + (0.951 + 0.309i)37-s + (−0.587 + 0.190i)39-s − 1.61·43-s + (−0.5 + 0.363i)47-s − 0.618·49-s + (0.587 + 0.809i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8635128577\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8635128577\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + 0.618T + T^{2} \) |
| 11 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.587 - 0.190i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.951 + 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.587 - 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-1.53 - 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.363 + 0.5i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−8.840289945623616185995603484929, −8.101744690611548006633112364858, −7.11865791980504650182398558762, −6.52623023642738938175352059933, −5.70673748637457216814895324191, −5.08421907892918753901910355842, −4.36391136422109662963592604527, −3.40705796161011591993973174261, −2.56797105004540820397226871596, −1.04167940902770683982657533805,
0.67883757351304709131047255957, 1.77112715521194536998217613180, 3.12934380897232612678322026610, 3.72543614723940592243025582855, 4.96299830368133065027202713188, 5.62241510746306114841708313192, 6.45872621711591649898604355649, 6.68560684437288558207663474071, 7.71380012491779061842935042825, 8.379219112727580198434890957639