L(s) = 1 | + 2-s + 4-s + (2.12 + 0.707i)5-s + (1 − i)7-s + 8-s + (2.12 + 0.707i)10-s + (1.58 + 1.58i)11-s − 3i·13-s + (1 − i)14-s + 16-s + (2.12 − 3.53i)17-s − 7.24i·19-s + (2.12 + 0.707i)20-s + (1.58 + 1.58i)22-s + (−2.82 + 2.82i)23-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (0.948 + 0.316i)5-s + (0.377 − 0.377i)7-s + 0.353·8-s + (0.670 + 0.223i)10-s + (0.478 + 0.478i)11-s − 0.832i·13-s + (0.267 − 0.267i)14-s + 0.250·16-s + (0.514 − 0.857i)17-s − 1.66i·19-s + (0.474 + 0.158i)20-s + (0.338 + 0.338i)22-s + (−0.589 + 0.589i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.405388618\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.405388618\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 - 0.707i)T \) |
| 17 | \( 1 + (-2.12 + 3.53i)T \) |
good | 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.58 - 1.58i)T + 11iT^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 19 | \( 1 + 7.24iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 - 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.707 - 0.707i)T - 29iT^{2} \) |
| 31 | \( 1 + (5.36 - 5.36i)T - 31iT^{2} \) |
| 37 | \( 1 + (-5.24 - 5.24i)T + 37iT^{2} \) |
| 41 | \( 1 + (4.41 + 4.41i)T + 41iT^{2} \) |
| 43 | \( 1 - 3.75T + 43T^{2} \) |
| 47 | \( 1 + 1.58iT - 47T^{2} \) |
| 53 | \( 1 - 3T + 53T^{2} \) |
| 59 | \( 1 - 12.8iT - 59T^{2} \) |
| 61 | \( 1 + (-6.12 - 6.12i)T + 61iT^{2} \) |
| 67 | \( 1 + 14.4iT - 67T^{2} \) |
| 71 | \( 1 + (3.70 - 3.70i)T - 71iT^{2} \) |
| 73 | \( 1 + (8.36 + 8.36i)T + 73iT^{2} \) |
| 79 | \( 1 + (0.242 + 0.242i)T + 79iT^{2} \) |
| 83 | \( 1 + 4.24T + 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 97 | \( 1 + (-0.121 - 0.121i)T + 97iT^{2} \) |
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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
−9.538663155551472021407709329116, −8.749227729146525009419809915019, −7.41529203953899936593765537743, −7.08508166287346664447671680877, −6.02514994802760319652996229107, −5.26976959158557264851916912952, −4.55692723871900871508865275860, −3.32187677505830848092071676884, −2.47418789283766730793330171467, −1.25180469779936044579153673938,
1.51404245521288187192414799228, 2.24381754400399942528565228330, 3.66879238783002312612685296806, 4.38466590741212737355676699864, 5.68491239660062480520615429613, 5.84772215565577164389876645236, 6.78803663622085398046391618680, 7.983752113194793445503738188737, 8.615775975282315733616836189088, 9.606786113355065035824520675471