L(s) = 1 | − i·3-s + 26i·7-s + 26·9-s + 45·11-s − 44i·13-s + 117i·17-s + 91·19-s + 26·21-s + 18i·23-s − 53i·27-s − 144·29-s + 26·31-s − 45i·33-s − 214i·37-s − 44·39-s + ⋯ |
L(s) = 1 | − 0.192i·3-s + 1.40i·7-s + 0.962·9-s + 1.23·11-s − 0.938i·13-s + 1.66i·17-s + 1.09·19-s + 0.270·21-s + 0.163i·23-s − 0.377i·27-s − 0.922·29-s + 0.150·31-s − 0.237i·33-s − 0.950i·37-s − 0.180·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.65916 + 0.391675i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65916 + 0.391675i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + iT - 27T^{2} \) |
| 7 | \( 1 - 26iT - 343T^{2} \) |
| 11 | \( 1 - 45T + 1.33e3T^{2} \) |
| 13 | \( 1 + 44iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 117iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 91T + 6.85e3T^{2} \) |
| 23 | \( 1 - 18iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 144T + 2.43e4T^{2} \) |
| 31 | \( 1 - 26T + 2.97e4T^{2} \) |
| 37 | \( 1 + 214iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 459T + 6.89e4T^{2} \) |
| 43 | \( 1 - 460iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 468iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 558iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 72T + 2.05e5T^{2} \) |
| 61 | \( 1 + 118T + 2.26e5T^{2} \) |
| 67 | \( 1 - 251iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 108T + 3.57e5T^{2} \) |
| 73 | \( 1 + 299iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 898T + 4.93e5T^{2} \) |
| 83 | \( 1 + 927iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 351T + 7.04e5T^{2} \) |
| 97 | \( 1 - 386iT - 9.12e5T^{2} \) |
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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
−13.23685780285049394962348753335, −12.42442482096761679933996178079, −11.57849437813299749432542718192, −10.13399282547714886175612445311, −9.099306185678696802390083690620, −7.985330032480269273994850694375, −6.53777620311412573802084483033, −5.41007124147762024486604492193, −3.63260068562254612262995119738, −1.69036601213490694776935665105,
1.20542120539560899129498037758, 3.70804106047148059952631965846, 4.74320459809182051351148795248, 6.79863395467540329750603402373, 7.38174822065435501609993085107, 9.233911830151405501417135977977, 9.949827041776075875339728072208, 11.21835019734493481088449009373, 12.12961248404188290857981544684, 13.72918646526290778297024060138