L(s) = 1 | − 4.47i·3-s + 5i·5-s − 31.3·7-s + 6.99·9-s − 8.94i·11-s − 62i·13-s + 22.3·15-s − 46·17-s − 107. i·19-s + 140i·21-s + 192.·23-s − 25·25-s − 152. i·27-s − 90i·29-s + 152.·31-s + ⋯ |
L(s) = 1 | − 0.860i·3-s + 0.447i·5-s − 1.69·7-s + 0.259·9-s − 0.245i·11-s − 1.32i·13-s + 0.384·15-s − 0.656·17-s − 1.29i·19-s + 1.45i·21-s + 1.74·23-s − 0.200·25-s − 1.08i·27-s − 0.576i·29-s + 0.880·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3761445709\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3761445709\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
good | 3 | \( 1 + 4.47iT - 27T^{2} \) |
| 7 | \( 1 + 31.3T + 343T^{2} \) |
| 11 | \( 1 + 8.94iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 62iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 46T + 4.91e3T^{2} \) |
| 19 | \( 1 + 107. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 192.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 90iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 214iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 10T + 6.89e4T^{2} \) |
| 43 | \( 1 + 67.0iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 398.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 678iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 411. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 250iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 49.1iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 366.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 522T + 3.89e5T^{2} \) |
| 79 | \( 1 + 876.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 380. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 970T + 7.04e5T^{2} \) |
| 97 | \( 1 + 934T + 9.12e5T^{2} \) |
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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
−8.805355208717398810455407199720, −7.80487213984748130902835536669, −6.85018687114563848646621162025, −6.65356687791172360875548144603, −5.68629735062512243416965916287, −4.44457021036536576879944503336, −3.01918008628723528847185945080, −2.77270845465095908462591800105, −1.02433737059245267682862169867, −0.10031340865744941966204890476,
1.50222520312761604120506180272, 2.95669779110963077305929985737, 3.87136145907700723040194633779, 4.52706114123846418432775803782, 5.55217778415788084856059570988, 6.66002748437146402496611828825, 7.04280366621862543695144092733, 8.490304260294749661648194961014, 9.292689749121940998907952504201, 9.640527528386427250088905975732