| L(s) = 1 | − 0.561·2-s − 7.68·4-s + 7·7-s + 8.80·8-s + 25.8·11-s + 15.5·13-s − 3.93·14-s + 56.5·16-s − 95.1·17-s − 143.·19-s − 14.4·22-s + 77.5·23-s − 8.73·26-s − 53.7·28-s + 204.·29-s − 40.6·31-s − 102.·32-s + 53.4·34-s − 95.6·37-s + 80.5·38-s − 24.7·41-s − 282.·43-s − 198.·44-s − 43.5·46-s + 257.·47-s + 49·49-s − 119.·52-s + ⋯ |
| L(s) = 1 | − 0.198·2-s − 0.960·4-s + 0.377·7-s + 0.389·8-s + 0.707·11-s + 0.331·13-s − 0.0750·14-s + 0.883·16-s − 1.35·17-s − 1.73·19-s − 0.140·22-s + 0.702·23-s − 0.0658·26-s − 0.363·28-s + 1.30·29-s − 0.235·31-s − 0.564·32-s + 0.269·34-s − 0.425·37-s + 0.343·38-s − 0.0941·41-s − 1.00·43-s − 0.679·44-s − 0.139·46-s + 0.798·47-s + 0.142·49-s − 0.318·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 0.561T + 8T^{2} \) |
| 11 | \( 1 - 25.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 15.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 95.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 143.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 77.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 204.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 95.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 24.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 282.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 257.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 257.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 651.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 451.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 832.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 174.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 47.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.16T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.49e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
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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
−8.689257424532438980298814428305, −8.230175594864463793772414311778, −6.97305647109596391661098159589, −6.34234235654288878296059844512, −5.18093353796941387762300031023, −4.40788446758935621230015993616, −3.79294837469130521737204776784, −2.34758180111414412280306954723, −1.17605116690899426035567815458, 0,
1.17605116690899426035567815458, 2.34758180111414412280306954723, 3.79294837469130521737204776784, 4.40788446758935621230015993616, 5.18093353796941387762300031023, 6.34234235654288878296059844512, 6.97305647109596391661098159589, 8.230175594864463793772414311778, 8.689257424532438980298814428305
