| L(s) = 1 | − 2-s − 2.23·3-s + 4-s + 0.618·5-s + 2.23·6-s − 2.85·7-s − 8-s + 2.00·9-s − 0.618·10-s − 2.23·12-s − 1.61·13-s + 2.85·14-s − 1.38·15-s + 16-s + 6.23·17-s − 2.00·18-s − 19-s + 0.618·20-s + 6.38·21-s − 0.236·23-s + 2.23·24-s − 4.61·25-s + 1.61·26-s + 2.23·27-s − 2.85·28-s + 10.4·29-s + 1.38·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.276·5-s + 0.912·6-s − 1.07·7-s − 0.353·8-s + 0.666·9-s − 0.195·10-s − 0.645·12-s − 0.448·13-s + 0.762·14-s − 0.356·15-s + 0.250·16-s + 1.51·17-s − 0.471·18-s − 0.229·19-s + 0.138·20-s + 1.39·21-s − 0.0492·23-s + 0.456·24-s − 0.923·25-s + 0.317·26-s + 0.430·27-s − 0.539·28-s + 1.94·29-s + 0.252·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4436135683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4436135683\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 - 6.23T + 17T^{2} \) |
| 23 | \( 1 + 0.236T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 7.85T + 37T^{2} \) |
| 41 | \( 1 + 4.09T + 41T^{2} \) |
| 43 | \( 1 + 3.70T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 - 5.94T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - 4.38T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 + 3.76T + 71T^{2} \) |
| 73 | \( 1 + 1.32T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 1.09T + 83T^{2} \) |
| 89 | \( 1 + 9.18T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−8.384128100991849732609244420164, −7.36487406471447669889341876599, −6.91171087307511925945520357482, −6.03653597334224017526757929125, −5.70255533127782000756365915216, −4.87835054082976535854434014287, −3.66165213990602140085641009450, −2.84291072239016416493754864037, −1.59023624936314229582331173761, −0.44397210129866027016585612795,
0.44397210129866027016585612795, 1.59023624936314229582331173761, 2.84291072239016416493754864037, 3.66165213990602140085641009450, 4.87835054082976535854434014287, 5.70255533127782000756365915216, 6.03653597334224017526757929125, 6.91171087307511925945520357482, 7.36487406471447669889341876599, 8.384128100991849732609244420164
