| L(s) = 1 | + 1.73i·3-s + 2·4-s + (2.5 + 0.866i)7-s − 2.99·9-s + 3.46i·12-s + 5.19i·13-s + 4·16-s − 8.66i·19-s + (−1.49 + 4.33i)21-s − 5.19i·27-s + (5 + 1.73i)28-s + 8.66i·31-s − 5.99·36-s − 10·37-s − 9·39-s + ⋯ |
| L(s) = 1 | + 0.999i·3-s + 4-s + (0.944 + 0.327i)7-s − 0.999·9-s + 0.999i·12-s + 1.44i·13-s + 16-s − 1.98i·19-s + (−0.327 + 0.944i)21-s − 0.999i·27-s + (0.944 + 0.327i)28-s + 1.55i·31-s − 0.999·36-s − 1.64·37-s − 1.44·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.54238 + 1.09800i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.54238 + 1.09800i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 8.66iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8.66iT - 61T^{2} \) |
| 67 | \( 1 + 5T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 13.8iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 19.0iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.10842878759503743210556032085, −10.37275353067317937034519186629, −9.145408195227827669094909749635, −8.634713223217658252321103660565, −7.32759021226586969853079106367, −6.48808743520627539616072849317, −5.25001449412478868326130679873, −4.49607333079499478568509703407, −3.10279454558850298002594662363, −1.92888964427847317577486867770,
1.24972451968203369021735875245, 2.34562082576932305220409514843, 3.63581623058477547447572411624, 5.46228399035234945665756082909, 6.03813736966547826216118988599, 7.28778582772320135952983078789, 7.82061418851378274455513592247, 8.449740893258303521461026460018, 10.13517445056906476310256448150, 10.77561790762563975980625846698
