| L(s) = 1 | + (−0.292 + 1.70i)3-s + (−1 − i)7-s + (−2.82 − i)9-s − 1.41i·11-s + (1.41 − 1.41i)17-s − 4i·19-s + (2 − 1.41i)21-s + (−2.82 − 2.82i)23-s + (2.53 − 4.53i)27-s + 7.07·29-s + 2·31-s + (2.41 + 0.414i)33-s + (−6 − 6i)37-s + 5.65i·41-s + (6 − 6i)43-s + ⋯ |
| L(s) = 1 | + (−0.169 + 0.985i)3-s + (−0.377 − 0.377i)7-s + (−0.942 − 0.333i)9-s − 0.426i·11-s + (0.342 − 0.342i)17-s − 0.917i·19-s + (0.436 − 0.308i)21-s + (−0.589 − 0.589i)23-s + (0.487 − 0.872i)27-s + 1.31·29-s + 0.359·31-s + (0.420 + 0.0721i)33-s + (−0.986 − 0.986i)37-s + 0.883i·41-s + (0.914 − 0.914i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109302240\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109302240\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.292 - 1.70i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (1 + i)T + 7iT^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 13iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 + 2.82i)T + 23iT^{2} \) |
| 29 | \( 1 - 7.07T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + (6 + 6i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 + (-6 + 6i)T - 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (2.82 + 2.82i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + (4 + 4i)T + 67iT^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (-5 + 5i)T - 73iT^{2} \) |
| 79 | \( 1 + 6iT - 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−9.781334180479654823515008619079, −8.911836419984599405008939111994, −8.256833532126280162106599968128, −7.06525891281992189520234477108, −6.23981948073651977075704086946, −5.30193416264812846710918154016, −4.45345377798473796243884228350, −3.54602120100338245685285433368, −2.60046458156208062329972334724, −0.52114991403707118090231917468,
1.30091233928686285794980887182, 2.43395813138501739073090722023, 3.51554869002160709638980914341, 4.85610762069170490932897964477, 5.89549984311615117020681657148, 6.41993687001169517417098489733, 7.42470702873921167196428049665, 8.086596366254903413841697401485, 8.885553014548806211303888361764, 9.887290878023623269791921938579
