L(s) = 1 | − 3·3-s − 7.52i·5-s + (4.86 − 17.8i)7-s + 9·9-s + 29.4i·11-s + 5.35i·13-s + 22.5i·15-s − 66.3i·17-s − 22.3·19-s + (−14.5 + 53.6i)21-s − 33.9i·23-s + 68.4·25-s − 27·27-s + 133.·29-s + 323.·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.672i·5-s + (0.262 − 0.964i)7-s + 0.333·9-s + 0.806i·11-s + 0.114i·13-s + 0.388i·15-s − 0.946i·17-s − 0.270·19-s + (−0.151 + 0.557i)21-s − 0.308i·23-s + 0.547·25-s − 0.192·27-s + 0.853·29-s + 1.87·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.591396526\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.591396526\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + (-4.86 + 17.8i)T \) |
good | 5 | \( 1 + 7.52iT - 125T^{2} \) |
| 11 | \( 1 - 29.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 5.35iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 66.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 22.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 33.9iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 133.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 323.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 120.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 140. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 19.0iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 376.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 441.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 241.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 130. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 627. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 808. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 417. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 214. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 639.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 686. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 103. iT - 9.12e5T^{2} \) |
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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
−8.994268768767099512522215137910, −8.202217314648113297329648832278, −7.18829430428545317957311544234, −6.73405746132215632131817339740, −5.51997270770620590431693672026, −4.61915728718129934827567887348, −4.24843478090826957558926320652, −2.70206541314125665420854480135, −1.31232507447489349344413108847, −0.48907393495055522748944554826,
1.03843903598833781471259177340, 2.37953788963474483980019547046, 3.27735791870576655059356371561, 4.47808576942380297012107999799, 5.45579507525349016678020269127, 6.19698357589171994327598300211, 6.75876042330701679250133316974, 8.013635782833024272724896498023, 8.545492921257936318392388724135, 9.509467793902934685375852432489