L(s) = 1 | + 2.44i·5-s + 2i·7-s + 4.89·11-s + 3.46·13-s − 4.24i·17-s − 6.92i·19-s + 8.48·23-s − 0.999·25-s − 7.34i·29-s + 2i·31-s − 4.89·35-s − 6.92·37-s + 4.24i·41-s + 8.48·47-s + 3·49-s + ⋯ |
L(s) = 1 | + 1.09i·5-s + 0.755i·7-s + 1.47·11-s + 0.960·13-s − 1.02i·17-s − 1.58i·19-s + 1.76·23-s − 0.199·25-s − 1.36i·29-s + 0.359i·31-s − 0.828·35-s − 1.13·37-s + 0.662i·41-s + 1.23·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.179475785\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.179475785\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.44iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 8.48T + 23T^{2} \) |
| 29 | \( 1 + 7.34iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + 6.92T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 2.44iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 14iT - 79T^{2} \) |
| 83 | \( 1 - 4.89T + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−9.060890347334159817886986013207, −8.563991486783831119517733026214, −7.18176539073575559584774012033, −6.86385441155677302328006573426, −6.13067662653973925922595847906, −5.17039990309429690474302921785, −4.16534758217647892315002082240, −3.12338176835940770015394716436, −2.52948184473363321141289681488, −1.05208011942309155525426782425,
1.09065120926770820063246310275, 1.55694744166573482409835677213, 3.55367544704249605263066157122, 3.90753085374511662244100508989, 4.90101863950603248879441320812, 5.79823705499227704859683804460, 6.59509333004898273289634208473, 7.36851190314382552730185948702, 8.422181833849579040636338950338, 8.840423343673042325686516617854