L(s) = 1 | + 1.41i·3-s − i·5-s − 4.24·7-s + 0.999·9-s − 5.65i·11-s − 2i·13-s + 1.41·15-s + 6·17-s − 2.82i·19-s − 6i·21-s + 7.07·23-s − 25-s + 5.65i·27-s − 4i·29-s − 2.82·31-s + ⋯ |
L(s) = 1 | + 0.816i·3-s − 0.447i·5-s − 1.60·7-s + 0.333·9-s − 1.70i·11-s − 0.554i·13-s + 0.365·15-s + 1.45·17-s − 0.648i·19-s − 1.30i·21-s + 1.47·23-s − 0.200·25-s + 1.08i·27-s − 0.742i·29-s − 0.508·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09696 - 0.454375i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09696 - 0.454375i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 1.41iT - 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 2.82iT - 59T^{2} \) |
| 61 | \( 1 + 14iT - 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−10.36926413967852164743846716464, −9.547626964815363223420711590115, −9.039040700517945874320302359160, −7.963424228184603107423239082070, −6.77848204638477211022921944776, −5.82232411586035505478478461304, −5.01126945023178015261143991850, −3.51306791888257889842289374559, −3.19568186379880682856062887964, −0.69360634084577674764769493279,
1.51534072924660585125836413196, 2.86764407793390778683761792912, 3.93747824915481641699339942293, 5.34612939172440329244279706201, 6.64955722366092072653421203520, 6.95263691217372191344450813372, 7.70348213657919961821157465824, 9.142449270544224216333849997726, 9.924022877947080583335173486520, 10.35400679362197395245451932633