L(s) = 1 | − 7·5-s − 11·7-s + 3·11-s + 11.3i·13-s + 17·17-s + 19·19-s + 2·23-s + 24·25-s − 39.5i·29-s + 5.65i·31-s + 77·35-s + 39.5i·37-s + 39.5i·41-s + 21·43-s − 5·47-s + ⋯ |
L(s) = 1 | − 1.40·5-s − 1.57·7-s + 0.272·11-s + 0.870i·13-s + 17-s + 19-s + 0.0869·23-s + 0.959·25-s − 1.36i·29-s + 0.182i·31-s + 2.19·35-s + 1.07i·37-s + 0.965i·41-s + 0.488·43-s − 0.106·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1630138767\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1630138767\) |
\(L(2)\) |
|
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 | \( 1 - 19T \) |
good | 5 | \( 1 + 7T + 25T^{2} \) |
| 7 | \( 1 + 11T + 49T^{2} \) |
| 11 | \( 1 - 3T + 121T^{2} \) |
| 13 | \( 1 - 11.3iT - 169T^{2} \) |
| 17 | \( 1 - 17T + 289T^{2} \) |
| 23 | \( 1 - 2T + 529T^{2} \) |
| 29 | \( 1 + 39.5iT - 841T^{2} \) |
| 31 | \( 1 - 5.65iT - 961T^{2} \) |
| 37 | \( 1 - 39.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 21T + 1.84e3T^{2} \) |
| 47 | \( 1 + 5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 5.65iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 23T + 3.72e3T^{2} \) |
| 67 | \( 1 - 39.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 39T + 5.32e3T^{2} \) |
| 79 | \( 1 + 96.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 169. iT - 9.40e3T^{2} \) |
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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
−9.177305583376216898726970618883, −8.081135826890479297695974816702, −7.59151813140525724586077676305, −6.72442401389623998453843864096, −6.20141626398805930106019693827, −5.05983668902382265487330444527, −4.03188017027254194821607328455, −3.51711908111293772617283926915, −2.73915680265906315362163159474, −1.05039645358483421031747458107,
0.05492748429264239096485275246, 0.967490205696825415777589730866, 2.83421843604645816279824663538, 3.45799986699934229368587296129, 3.96103636500002537358321257262, 5.24965394737118642965173912021, 5.91529865222587186567521362696, 7.04680744191219891712001137407, 7.35407086765823846071304169297, 8.208850659089930954135162480977