L(s) = 1 | + i·7-s + i·9-s + (1 − i)11-s − i·25-s + (−1 − i)29-s + (−1 + i)37-s + (−1 + i)43-s − 49-s + (1 − i)53-s − 63-s + (−1 − i)67-s − 2i·71-s + (1 + i)77-s − 81-s + (1 + i)99-s + ⋯ |
L(s) = 1 | + i·7-s + i·9-s + (1 − i)11-s − i·25-s + (−1 − i)29-s + (−1 + i)37-s + (−1 + i)43-s − 49-s + (1 − i)53-s − 63-s + (−1 − i)67-s − 2i·71-s + (1 + i)77-s − 81-s + (1 + i)99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8788556582\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8788556582\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (-1 + i)T - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (1 + i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 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
−11.51476265427998245659871050178, −10.53743902176011287950925936075, −9.498060544688162463117669866862, −8.601569094420962261274312203669, −7.949616212203672260157876602020, −6.56223349854719080291047130507, −5.75676923305405776818062078547, −4.69881068639797943744084297627, −3.30998008914652883245525121937, −1.98122159041581201698815890823,
1.51577420840759217357557168397, 3.50179385784055813431133562100, 4.23739463083575739851460564498, 5.59279624830757051015126000143, 7.00291672716090395172122455978, 7.15390199970132659411349154919, 8.741116334776820343815578132825, 9.474016314938493682713111387006, 10.29676061488334960397887815207, 11.28453849739738996049913712636