L(s) = 1 | + 0.253·2-s − 1.93·4-s − 2.92i·5-s + 2.71i·7-s − 0.998·8-s − 0.741i·10-s + 2.85i·13-s + 0.689i·14-s + 3.61·16-s − 4.46·17-s + 1.91i·19-s + 5.65i·20-s + 0.816i·23-s − 3.53·25-s + 0.724i·26-s + ⋯ |
L(s) = 1 | + 0.179·2-s − 0.967·4-s − 1.30i·5-s + 1.02i·7-s − 0.352·8-s − 0.234i·10-s + 0.792i·13-s + 0.184i·14-s + 0.904·16-s − 1.08·17-s + 0.438i·19-s + 1.26i·20-s + 0.170i·23-s − 0.707·25-s + 0.142i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.061798827\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.061798827\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 0.253T + 2T^{2} \) |
| 5 | \( 1 + 2.92iT - 5T^{2} \) |
| 7 | \( 1 - 2.71iT - 7T^{2} \) |
| 13 | \( 1 - 2.85iT - 13T^{2} \) |
| 17 | \( 1 + 4.46T + 17T^{2} \) |
| 19 | \( 1 - 1.91iT - 19T^{2} \) |
| 23 | \( 1 - 0.816iT - 23T^{2} \) |
| 29 | \( 1 - 9.54T + 29T^{2} \) |
| 31 | \( 1 + 5.98T + 31T^{2} \) |
| 37 | \( 1 - 5.94T + 37T^{2} \) |
| 41 | \( 1 - 8.43T + 41T^{2} \) |
| 43 | \( 1 - 11.8iT - 43T^{2} \) |
| 47 | \( 1 - 7.71iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 - 0.0932iT - 59T^{2} \) |
| 61 | \( 1 - 2.14iT - 61T^{2} \) |
| 67 | \( 1 + 1.40T + 67T^{2} \) |
| 71 | \( 1 + 3.60iT - 71T^{2} \) |
| 73 | \( 1 + 1.75iT - 73T^{2} \) |
| 79 | \( 1 - 7.13iT - 79T^{2} \) |
| 83 | \( 1 - 7.83T + 83T^{2} \) |
| 89 | \( 1 + 2.06iT - 89T^{2} \) |
| 97 | \( 1 - 1.67T + 97T^{2} \) |
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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
−9.447412535715943618426272228061, −9.256326115168553880397511011843, −8.554162835837617604665638094774, −7.85193587416738381439118607530, −6.33862821889183677875879596692, −5.60389039919312366982166960719, −4.63769781252510577749248811894, −4.25831900038581181926724342006, −2.68073281789993518962117718888, −1.20984153297387632775226526441,
0.52651304948994397670357587418, 2.56028238562377240262453770252, 3.57911002508523718450098004071, 4.33834923233688202308333260236, 5.37875087454917710379005477980, 6.50735585856415705352205759272, 7.14096657028307398648607942997, 8.048504852810878907887507167339, 8.928348616585897889177228078860, 9.918358935470082578808637619141