| L(s) = 1 | + (−4.89 − 4.89i)7-s + 15·11-s + (−2.44 + 2.44i)13-s + (3.67 + 3.67i)17-s + 17i·19-s + (7.34 − 7.34i)23-s − 42i·29-s − 14·31-s + (−29.3 − 29.3i)37-s + 39·41-s + (24.4 − 24.4i)43-s + (−51.4 − 51.4i)47-s − 1.00i·49-s + (66.1 − 66.1i)53-s + 6i·59-s + ⋯ |
| L(s) = 1 | + (−0.699 − 0.699i)7-s + 1.36·11-s + (−0.188 + 0.188i)13-s + (0.216 + 0.216i)17-s + 0.894i·19-s + (0.319 − 0.319i)23-s − 1.44i·29-s − 0.451·31-s + (−0.794 − 0.794i)37-s + 0.951·41-s + (0.569 − 0.569i)43-s + (−1.09 − 1.09i)47-s − 0.0204i·49-s + (1.24 − 1.24i)53-s + 0.101i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.638656160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.638656160\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (4.89 + 4.89i)T + 49iT^{2} \) |
| 11 | \( 1 - 15T + 121T^{2} \) |
| 13 | \( 1 + (2.44 - 2.44i)T - 169iT^{2} \) |
| 17 | \( 1 + (-3.67 - 3.67i)T + 289iT^{2} \) |
| 19 | \( 1 - 17iT - 361T^{2} \) |
| 23 | \( 1 + (-7.34 + 7.34i)T - 529iT^{2} \) |
| 29 | \( 1 + 42iT - 841T^{2} \) |
| 31 | \( 1 + 14T + 961T^{2} \) |
| 37 | \( 1 + (29.3 + 29.3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 39T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-24.4 + 24.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (51.4 + 51.4i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-66.1 + 66.1i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 92T + 3.72e3T^{2} \) |
| 67 | \( 1 + (35.5 + 35.5i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 102T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-3.67 + 3.67i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 104iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (47.7 - 47.7i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 87iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-93.0 - 93.0i)T + 9.40e3iT^{2} \) |
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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
−9.789590489163760920561496417792, −9.050005734916296794658295135595, −8.076161577766651644509798330440, −7.04836845450508109899656114592, −6.46948369975955381317784862760, −5.47197300825134159297886221615, −4.06340480736354985610280641399, −3.62277918610073906154710915249, −2.02975089262195843261809410429, −0.61436178852592983934516491072,
1.17248707323914511741951684708, 2.67223944949086130621672342583, 3.59351895249567992140613370941, 4.78325383898664299619734113714, 5.78962836562753167008162879637, 6.64982625733985395534261601427, 7.35004782007268651940320670818, 8.670664784581418230074005667417, 9.187213750715146546618645748408, 9.857340679050375873774914946247
