L(s) = 1 | + (−0.546 + 0.837i)2-s − 1.93i·3-s + (−0.401 − 0.915i)4-s − 1.99i·5-s + (1.62 + 1.06i)6-s + (0.986 + 0.164i)8-s − 2.75·9-s + (1.66 + 1.09i)10-s − 1.22i·11-s + (−1.77 + 0.778i)12-s − 3.86·15-s + (−0.677 + 0.735i)16-s + 1.35·17-s + (1.50 − 2.30i)18-s + (−1.82 + 0.800i)20-s + ⋯ |
L(s) = 1 | + (−0.546 + 0.837i)2-s − 1.93i·3-s + (−0.401 − 0.915i)4-s − 1.99i·5-s + (1.62 + 1.06i)6-s + (0.986 + 0.164i)8-s − 2.75·9-s + (1.66 + 1.09i)10-s − 1.22i·11-s + (−1.77 + 0.778i)12-s − 3.86·15-s + (−0.677 + 0.735i)16-s + 1.35·17-s + (1.50 − 2.30i)18-s + (−1.82 + 0.800i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7856218416\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7856218416\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.546 - 0.837i)T \) |
| 359 | \( 1 + T \) |
good | 3 | \( 1 + 1.93iT - T^{2} \) |
| 5 | \( 1 + 1.99iT - T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.22iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - 1.35T + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + 0.803T + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.649iT - T^{2} \) |
| 41 | \( 1 - 1.89T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.75T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.09T + T^{2} \) |
| 79 | \( 1 - 0.165T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.394770098834052542121572204985, −7.80176944852138844125937946742, −7.41132551515169116485681338536, −6.09767631446252617883313192729, −5.79655412386246138485407838076, −5.24320692851039369281427157890, −3.91380092434425878679202301547, −2.24873905499384032082559976584, −1.13800666926895219897710244748, −0.70622560025376278101023908869,
2.34891349665473470719827473796, 2.89363184291206467085385489010, 3.80117252747448064583199042420, 4.15637864088411236725842103814, 5.33359209871276042139587235884, 6.23587487552907385031388872016, 7.40378962676641564189608334734, 7.88697790970117633242176203627, 9.039402078923596944761513718481, 9.801713776083383442778785824340