| L(s) = 1 | + i·2-s − 4-s + 3i·5-s + i·7-s − i·8-s − 3·10-s + 6i·11-s − 14-s + 16-s − 3·17-s + 2i·19-s − 3i·20-s − 6·22-s − 4·25-s − i·28-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.34i·5-s + 0.377i·7-s − 0.353i·8-s − 0.948·10-s + 1.80i·11-s − 0.267·14-s + 0.250·16-s − 0.727·17-s + 0.458i·19-s − 0.670i·20-s − 1.27·22-s − 0.800·25-s − 0.188i·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.050796082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.050796082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3iT - 5T^{2} \) |
| 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 - 6iT - 11T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 - 3iT - 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−9.262264701549283183177122809711, −8.282576032456469430468124762169, −7.43746914954978508500396629293, −7.02733391773857010169922380208, −6.36858203234166590541429798030, −5.55308940778784518244614875905, −4.58623828978365290479165402455, −3.83764363317400707663408299826, −2.69531386057586455688469041846, −1.87964288525549116665171715554,
0.35677973148508813724090851998, 1.14090294087049871764155079724, 2.35561554234582204420478265706, 3.52516492586423539505627870980, 4.13730561755027515223036812509, 5.13055452842530545736592931516, 5.62897839384366436669934534113, 6.65156348237595339437632570520, 7.75942261759010038385503437478, 8.497714510557332281221310784655
