| L(s) = 1 | + 2-s + 4-s − 5-s + 5.14i·7-s + 8-s − 10-s + 3.05i·11-s + 5.39·13-s + 5.14i·14-s + 16-s + 1.31i·17-s − 7.59i·19-s − 20-s + 3.05i·22-s − 0.929i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.94i·7-s + 0.353·8-s − 0.316·10-s + 0.919i·11-s + 1.49·13-s + 1.37i·14-s + 0.250·16-s + 0.318i·17-s − 1.74i·19-s − 0.223·20-s + 0.650i·22-s − 0.193i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.808465700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.808465700\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 + (-4.93 + 4.32i)T \) |
| good | 7 | \( 1 - 5.14iT - 7T^{2} \) |
| 11 | \( 1 - 3.05iT - 11T^{2} \) |
| 13 | \( 1 - 5.39T + 13T^{2} \) |
| 17 | \( 1 - 1.31iT - 17T^{2} \) |
| 19 | \( 1 + 7.59iT - 19T^{2} \) |
| 23 | \( 1 + 0.929iT - 23T^{2} \) |
| 29 | \( 1 + 1.08T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 - 4.57iT - 37T^{2} \) |
| 41 | \( 1 - 6.33iT - 41T^{2} \) |
| 47 | \( 1 - 4.95iT - 47T^{2} \) |
| 53 | \( 1 - 12.0iT - 53T^{2} \) |
| 59 | \( 1 - 14.5iT - 59T^{2} \) |
| 61 | \( 1 + 9.02iT - 61T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 + 9.21T + 71T^{2} \) |
| 73 | \( 1 - 5.35iT - 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 6.72iT - 83T^{2} \) |
| 89 | \( 1 + 2.64T + 89T^{2} \) |
| 97 | \( 1 - 13.9T + 97T^{2} \) |
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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
−8.763547252201360275550828105321, −7.990489050906916972626150253004, −7.04720250910967136168363150816, −6.22855779870018373141234858744, −5.79475398258408299276849405549, −4.79739819046932357432131642922, −4.30706181418559129333497050594, −3.02624564962947150624473394446, −2.57911424959512489732140614854, −1.41852787433429999601798384771,
0.67208054745138856060727401161, 1.55334974972797189469132779535, 3.19171635922180749164150537017, 3.83736816790104033260688774042, 4.09252078728604304780523688027, 5.25747421262361130539011738968, 6.13654600693398606993072826487, 6.68078150056427326847304507583, 7.60682025478695794138644937962, 8.019540832905207875743606060075
