L(s) = 1 | + (0.707 − 0.707i)5-s + (1 − i)13-s − 1.00i·25-s − 1.41·29-s + (1 + i)37-s − 1.41i·41-s + i·49-s − 1.41i·65-s + (1 − i)73-s − 1.41·89-s + (1 + i)97-s − 1.41i·101-s + (1.41 + 1.41i)113-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)5-s + (1 − i)13-s − 1.00i·25-s − 1.41·29-s + (1 + i)37-s − 1.41i·41-s + i·49-s − 1.41i·65-s + (1 − i)73-s − 1.41·89-s + (1 + i)97-s − 1.41i·101-s + (1.41 + 1.41i)113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.437834302\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.437834302\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-1 + i)T - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1 + i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−8.885990131864725787340277399671, −8.181028068706345065031619576731, −7.48082322806505787024127677068, −6.35759006020709001070846881534, −5.78296567434150685505480316884, −5.12146484956554561611566709718, −4.14549755270071257599407314644, −3.21236202764935327546139350673, −2.08076936946938008534206277094, −1.00092017584885616291317561656,
1.51076933828162805237097319205, 2.39562164731800130786786219744, 3.46909484873863335559908819142, 4.23006908493572465921853690793, 5.38154692373228313663656558960, 6.08448862648407910074860592399, 6.70664653852982943578260943930, 7.44181326082767250668674177415, 8.349363461794503287421435372998, 9.220758420086748418565222567411