| L(s) = 1 | + (1.65 + 0.5i)3-s + 2·4-s + (2.5 + 1.65i)9-s − 3.31i·11-s + (3.31 + i)12-s + 4·16-s − 3.31·23-s + (3.31 + 4i)27-s + 5·31-s + (1.65 − 5.5i)33-s + (5 + 3.31i)36-s + 7i·37-s − 6.63i·44-s − 6.63·47-s + (6.63 + 2i)48-s − 7·49-s + ⋯ |
| L(s) = 1 | + (0.957 + 0.288i)3-s + 4-s + (0.833 + 0.552i)9-s − 1.00i·11-s + (0.957 + 0.288i)12-s + 16-s − 0.691·23-s + (0.638 + 0.769i)27-s + 0.898·31-s + (0.288 − 0.957i)33-s + (0.833 + 0.552i)36-s + 1.15i·37-s − 1.00i·44-s − 0.967·47-s + (0.957 + 0.288i)48-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.72514 + 0.233301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.72514 + 0.233301i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.65 - 0.5i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 3.31iT \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 - 7iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 6.63T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + 3.31iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13iT - 67T^{2} \) |
| 71 | \( 1 + 16.5iT - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 16.5iT - 89T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
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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
−10.20324108571103045416987134410, −9.483127877462646081595890727463, −8.294217281154357696975339044036, −7.965941830353378493519561021518, −6.81480510089470644901508748090, −6.04905093985603838217642726713, −4.78111757047763281471916857862, −3.50000704393480279798665514133, −2.80102582938657671223557221847, −1.57190933946848588697407658509,
1.59984828419113111853503225789, 2.48125306556065924330018349575, 3.52830324714720463326664137651, 4.68458176103516922734677433256, 6.12028795378701784533580220530, 6.89121559159717366734855349147, 7.65449478520807057916074678743, 8.290404999651457646172462531856, 9.481930903917713399555521557571, 10.04177083141962483398564805703
