L(s) = 1 | − 3-s + 5-s − 2.52·7-s + 9-s − 3.10·11-s + 6.72·13-s − 15-s − 2.57·17-s − 19-s + 2.52·21-s − 4.57·23-s + 25-s − 27-s − 1.10·29-s + 7.83·31-s + 3.10·33-s − 2.52·35-s − 4.52·37-s − 6.72·39-s + 6.15·41-s − 7.68·43-s + 45-s − 3.42·47-s − 0.627·49-s + 2.57·51-s + 2.57·53-s − 3.10·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.954·7-s + 0.333·9-s − 0.935·11-s + 1.86·13-s − 0.258·15-s − 0.625·17-s − 0.229·19-s + 0.550·21-s − 0.954·23-s + 0.200·25-s − 0.192·27-s − 0.204·29-s + 1.40·31-s + 0.540·33-s − 0.426·35-s − 0.743·37-s − 1.07·39-s + 0.960·41-s − 1.17·43-s + 0.149·45-s − 0.499·47-s − 0.0896·49-s + 0.361·51-s + 0.354·53-s − 0.418·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 + 3.10T + 11T^{2} \) |
| 13 | \( 1 - 6.72T + 13T^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 23 | \( 1 + 4.57T + 23T^{2} \) |
| 29 | \( 1 + 1.10T + 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 + 4.52T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 + 7.68T + 43T^{2} \) |
| 47 | \( 1 + 3.42T + 47T^{2} \) |
| 53 | \( 1 - 2.57T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 8.41T + 67T^{2} \) |
| 71 | \( 1 - 9.45T + 71T^{2} \) |
| 73 | \( 1 + 9.25T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−8.610750011138441833014484071164, −7.963354191479341623732652322253, −6.77230316278390298223509126641, −6.23854812555862474028484122781, −5.71395450387227694597263214000, −4.65654298619858796014788257304, −3.70376367865501333865061227317, −2.75046929565853231880203129369, −1.48291588483107598247836553906, 0,
1.48291588483107598247836553906, 2.75046929565853231880203129369, 3.70376367865501333865061227317, 4.65654298619858796014788257304, 5.71395450387227694597263214000, 6.23854812555862474028484122781, 6.77230316278390298223509126641, 7.963354191479341623732652322253, 8.610750011138441833014484071164