L(s) = 1 | + 6i·3-s − 5i·5-s + 34·7-s − 9·9-s + 16i·11-s − 58i·13-s + 30·15-s − 70·17-s − 4i·19-s + 204i·21-s + 134·23-s − 25·25-s + 108i·27-s + 242i·29-s + 100·31-s + ⋯ |
L(s) = 1 | + 1.15i·3-s − 0.447i·5-s + 1.83·7-s − 0.333·9-s + 0.438i·11-s − 1.23i·13-s + 0.516·15-s − 0.998·17-s − 0.0482i·19-s + 2.11i·21-s + 1.21·23-s − 0.200·25-s + 0.769i·27-s + 1.54i·29-s + 0.579·31-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
\(L(2)\) |
\(\approx\) |
\(2.913247195\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.913247195\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 6iT - 27T^{2} \) |
| 7 | \( 1 - 34T + 343T^{2} \) |
| 11 | \( 1 - 16iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 58iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 70T + 4.91e3T^{2} \) |
| 19 | \( 1 + 4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 134T + 1.21e4T^{2} \) |
| 29 | \( 1 - 242iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 100T + 2.97e4T^{2} \) |
| 37 | \( 1 + 438iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 138T + 6.89e4T^{2} \) |
| 43 | \( 1 - 178iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 22T + 1.03e5T^{2} \) |
| 53 | \( 1 - 162iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 268iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 250iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 422iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 852T + 3.57e5T^{2} \) |
| 73 | \( 1 + 306T + 3.89e5T^{2} \) |
| 79 | \( 1 + 456T + 4.93e5T^{2} \) |
| 83 | \( 1 + 434iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 726T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.37e3T + 9.12e5T^{2} \) |
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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
−9.239100361430189529722606322297, −8.777935974706803563534767402723, −7.895395184816386569843687363197, −7.15151580727155384064562646661, −5.65119151514297760264601478650, −4.82089375557884177803052948610, −4.64415048540858359589040417274, −3.45507706425198813174685781113, −2.10082722876539966745034798149, −0.924150554144501769831617238809,
0.905588982788223636322414048754, 1.80695137662523594493475965700, 2.53649711330345309035095437828, 4.19283627668087246014027380717, 4.87016605719736532203805294925, 6.11265740625908952219023256737, 6.83969770964335512117670545130, 7.51317087682150145388882948578, 8.289037458492586022701750380463, 8.860509625885330217548384967013