L(s) = 1 | + 4.01·5-s + 10.9·7-s + 9.36·11-s + 15.4i·13-s + 13.8·17-s + (18.2 − 5.22i)19-s + 35.0·23-s − 8.88·25-s − 25.0i·29-s − 20.4i·31-s + 43.8·35-s + 51.6i·37-s + 52.7i·41-s − 23.7·43-s − 4.83·47-s + ⋯ |
L(s) = 1 | + 0.802·5-s + 1.55·7-s + 0.851·11-s + 1.18i·13-s + 0.812·17-s + (0.961 − 0.274i)19-s + 1.52·23-s − 0.355·25-s − 0.865i·29-s − 0.660i·31-s + 1.25·35-s + 1.39i·37-s + 1.28i·41-s − 0.552·43-s − 0.102·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.961 - 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.739656057\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.739656057\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-18.2 + 5.22i)T \) |
good | 5 | \( 1 - 4.01T + 25T^{2} \) |
| 7 | \( 1 - 10.9T + 49T^{2} \) |
| 11 | \( 1 - 9.36T + 121T^{2} \) |
| 13 | \( 1 - 15.4iT - 169T^{2} \) |
| 17 | \( 1 - 13.8T + 289T^{2} \) |
| 23 | \( 1 - 35.0T + 529T^{2} \) |
| 29 | \( 1 + 25.0iT - 841T^{2} \) |
| 31 | \( 1 + 20.4iT - 961T^{2} \) |
| 37 | \( 1 - 51.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 52.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 4.83T + 2.20e3T^{2} \) |
| 53 | \( 1 + 70.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 81.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 12.6T + 3.72e3T^{2} \) |
| 67 | \( 1 - 74.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 52.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 119.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 118. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 0.221T + 6.88e3T^{2} \) |
| 89 | \( 1 + 77.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 104. iT - 9.40e3T^{2} \) |
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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
−8.637110445459408359563301264482, −8.020572298914415475861304409884, −7.12182291785125471247602008273, −6.43400851866223372287802918398, −5.46502050822114535323760683112, −4.85265203516863798606867621574, −4.06344897103910891833844746273, −2.84409820671548975817261436802, −1.68592966361625162296916359264, −1.21047699156466606806829703097,
1.05066624438220007759474270883, 1.61621669151163732167682861734, 2.84829407021279241899146337964, 3.78223052722557401354671351470, 5.02136604393758223246567074830, 5.35252091463320025232849737111, 6.14801213998515199625269463115, 7.42794440348886929625188435970, 7.62389453854069913797435776937, 8.842458645193966671577938572950