L(s) = 1 | + 3i·3-s − 4.35·5-s + (−7.09 − 17.1i)7-s − 9·9-s − 43.1·11-s + 26.5·13-s − 13.0i·15-s + 76.6i·17-s + 58.7i·19-s + (51.3 − 21.2i)21-s − 0.747i·23-s − 106.·25-s − 27i·27-s + 64.9i·29-s − 165.·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.389·5-s + (−0.382 − 0.923i)7-s − 0.333·9-s − 1.18·11-s + 0.565·13-s − 0.224i·15-s + 1.09i·17-s + 0.709i·19-s + (0.533 − 0.221i)21-s − 0.00677i·23-s − 0.848·25-s − 0.192i·27-s + 0.416i·29-s − 0.958·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.382i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.924 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.104323765\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.104323765\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| 7 | \( 1 + (7.09 + 17.1i)T \) |
good | 5 | \( 1 + 4.35T + 125T^{2} \) |
| 11 | \( 1 + 43.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 58.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 0.747iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 64.9iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 165.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 389. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 290. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 32.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 368.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 588. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 471. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 448.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 707.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 997. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.22e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 160. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.00e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 1.13e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 729. iT - 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.270643373479357081013394285124, −8.212592648681428133890998696831, −7.79405545422883164065896286777, −6.71878973411671331384005745755, −5.81289622207785225668824944257, −4.90701355366114509349544373839, −3.81944611521132811766063689532, −3.41906160002968610358577798447, −1.90374072065940961167408807923, −0.39326559086653509901839540411,
0.65603037894305478207536897361, 2.26030615875316032392765015142, 2.87117622493573620048817832472, 4.10700511668529860650724788537, 5.38826237095172200114526091661, 5.81953048194391144276017221213, 7.00822234630324591065556414078, 7.59970557354750042404858477908, 8.474027362478072772906736309877, 9.163634358409797725801064458095