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
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
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
−8.616971778128198817178950251433, −7.57706667744623064729707470650, −6.98373213895333664042005424253, −6.11010763630933158356187288671, −5.59919496055218049318046503027, −4.04536217251803621567822523093, −3.12087215124242464614403202141, −2.10398492204946148474757201630, −0.72432522782222773782146356481, 0,
2.11438454019595056294603807452, 3.19845965599820779294171939540, 4.03380857909281170993181713376, 4.61794446749442080352496139828, 6.09372016685551736636052756157, 6.51846868918154285693602356426, 7.40462246896954233825100931045, 8.724008917527551254855206246271, 9.587506912426515772653170542610