| L(s) = 1 | − 0.717i·5-s + 2.44·7-s + 4.24i·11-s − 4.18i·13-s − 3·17-s − 2.24i·19-s − 5.91·23-s + 4.48·25-s − 0.717i·29-s − 1.43·31-s − 1.75i·35-s − 2.74i·37-s + 8.48·41-s − 3.75i·43-s + 1.43·47-s + ⋯ |
| L(s) = 1 | − 0.320i·5-s + 0.925·7-s + 1.27i·11-s − 1.15i·13-s − 0.727·17-s − 0.514i·19-s − 1.23·23-s + 0.897·25-s − 0.133i·29-s − 0.257·31-s − 0.297i·35-s − 0.451i·37-s + 1.32·41-s − 0.572i·43-s + 0.209·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.776334684\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.776334684\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.717iT - 5T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 13 | \( 1 + 4.18iT - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + 2.24iT - 19T^{2} \) |
| 23 | \( 1 + 5.91T + 23T^{2} \) |
| 29 | \( 1 + 0.717iT - 29T^{2} \) |
| 31 | \( 1 + 1.43T + 31T^{2} \) |
| 37 | \( 1 + 2.74iT - 37T^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + 3.75iT - 43T^{2} \) |
| 47 | \( 1 - 1.43T + 47T^{2} \) |
| 53 | \( 1 + 11.8iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 9.08iT - 61T^{2} \) |
| 67 | \( 1 + 6.24iT - 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 9.48T + 73T^{2} \) |
| 79 | \( 1 - 1.01T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 0.514T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−8.053068061355024448452166351117, −7.45120966829100017299960107311, −6.72475021173546604486620406548, −5.78978307204469704033458350294, −4.97847844322860926503796474597, −4.56705584898159750151061169039, −3.64748922729936480040938290412, −2.43544477254838548524136277678, −1.79088255437501525458862951746, −0.50255826349988131354321547918,
1.11032638475170135034610206838, 2.08247882936421823221234949105, 2.97978618509470737151052790679, 4.04472476373572716376347718028, 4.52613094170142496431962886788, 5.58228511608505636863571793882, 6.17399621042739571413286841681, 6.90390839198886665234767628703, 7.72357608369025310161244444323, 8.384354373579537422891233770251
