L(s) = 1 | + 1.66i·2-s + 1.25i·3-s − 0.769·4-s + (−1.73 − 1.41i)5-s − 2.09·6-s − 4.13i·7-s + 2.04i·8-s + 1.41·9-s + (2.34 − 2.88i)10-s + 11-s − 0.969i·12-s + 3.80i·13-s + 6.88·14-s + (1.77 − 2.18i)15-s − 4.94·16-s + 2.64i·17-s + ⋯ |
L(s) = 1 | + 1.17i·2-s + 0.726i·3-s − 0.384·4-s + (−0.775 − 0.631i)5-s − 0.855·6-s − 1.56i·7-s + 0.723i·8-s + 0.471·9-s + (0.742 − 0.912i)10-s + 0.301·11-s − 0.279i·12-s + 1.05i·13-s + 1.83·14-s + (0.458 − 0.563i)15-s − 1.23·16-s + 0.640i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.488335869\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.488335869\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (1.73 + 1.41i)T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 1.66iT - 2T^{2} \) |
| 3 | \( 1 - 1.25iT - 3T^{2} \) |
| 7 | \( 1 + 4.13iT - 7T^{2} \) |
| 13 | \( 1 - 3.80iT - 13T^{2} \) |
| 17 | \( 1 - 2.64iT - 17T^{2} \) |
| 23 | \( 1 - 5.58iT - 23T^{2} \) |
| 29 | \( 1 - 6.34T + 29T^{2} \) |
| 31 | \( 1 - 7.53T + 31T^{2} \) |
| 37 | \( 1 + 3.00iT - 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 - 10.6iT - 43T^{2} \) |
| 47 | \( 1 - 6.54iT - 47T^{2} \) |
| 53 | \( 1 - 7.14iT - 53T^{2} \) |
| 59 | \( 1 - 7.34T + 59T^{2} \) |
| 61 | \( 1 - 4.72T + 61T^{2} \) |
| 67 | \( 1 + 7.94iT - 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 + 16.9iT - 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 + 1.51iT - 83T^{2} \) |
| 89 | \( 1 + 8.71T + 89T^{2} \) |
| 97 | \( 1 + 6.68iT - 97T^{2} \) |
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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
−10.10872900278963423579240946920, −9.295825699241677668499653836482, −8.363931769737181125669171461155, −7.63263484949713333271526739895, −6.98926354735683775514377266403, −6.22085292663170296300524665763, −4.73478433951335898172050631458, −4.45970990545250132480279673526, −3.56244535616931367807418036780, −1.34704184935520394758642339867,
0.75283742491721925025652966303, 2.33135465545683175860173256092, 2.76842446508186296094220019672, 3.91726396689457831059497784965, 5.12870954419923321497237968143, 6.50819557603741453865120615117, 6.88448324451136290406528687606, 8.156535311840288619786838717250, 8.674327854218106561413015730918, 10.04008466300706118561259255048